Yes, a priest can give a mass in American Sign Language (ASL)

Fr. Edward McNamara wrote an extensive article for the organization ZENIT about the validity of sign language at mass, specifically using sign language during the consecration and not vocally pronouncing the words.

Sign language is certainly permitted for the liturgy

First, Fr. McNamara first answers the general question of whether or not sign language can be used in liturgy, and he cites Notitiae, the official magazine of the Vatican Congregation for Divine Worship and the Discipline of the Sacraments.

Whether the language that is called « gestural » (sign-language) can be employed in the celebration of the liturgy for the deaf? In the affirmative. For it is the only system by which the deaf can truly actively participate in the sacred liturgy. As a matter of fact, to certain Conferences of Bishops that asked, the Most Blessed Father recently (14 Dec. 1965) kindly granted that the language that is called « gestural » can be employed in the celebration of the liturgy for the deaf, whenever a pastoral reason suggests it, in all the parts which are said in the vernacular language. The celebration can be ordered like this: The readings are conveyed to the congregation through signs. With respect to participation in the other parts which regard the people: a) what is pronounced by the celebrant alone, he himself at the same time pronounces the words and likewise conveys them by gestures; the people respond with gestures;

b) in the parts which are to be said together by the celebrant and the people, e.g., the Gloria, the Credo, the Sanctus-Benedictus, the Agnus Dei, etc., the faithful follow the signs of the celebrant through gestures. —Notitiae 2 (1966): 30-31 no. 95

There does not appear to be an official statement explicitly addressing validity during consecration, but consensus among canonists is that it is valid

However, it does not answer our reader’s precise question regarding validity, and, although there have been some good theological reflections on this topic and some serious canonical studies, I have been unable to find a definitive official declaration that would close the debate. Renowned canonist Dr. Edward Peters has published several studies on this topic and proposes a positive conclusion with respect to the sacramental validity of sign language. There are other theological studies which reach similar conclusions from the dogmatic standpoint. Unfortunately, most of these studies do not appear to be available online.

The Dr. Edward Peters he mentions maintains "A Bibliography of Deaf Catholic Sources and Studies", which include some of his own works that I believe Fr. McNamara is referencing.

What other evidence is there that this is permitted?

Fr. McNamara states that dioceses in the United States have dedicated Deaf Ministries that celebrate mass in American Sign Language, and defers to their authority in being able to do so. Beyond that, he gives several reasons that he concurs with Dr. Edward Peters' opinion:

Deaf priests have been ordained since 1977, and some of them are unable to verbally speak

I personally reached out to the coordinator of the Deaf Ministry in the Archdiocese of Baltimore about this question, and he cited this point as one of the pieces of evidence that it was valid:

For your information, there are more than 10 Deaf men who have been ordained priests. In fact, one was ordained by St. John Paul II himself. I am not a canon lawyer and cannot address all of the fine details related to canon law. However, I think we would have to assume that priests would not be ordained if it was known that they would be unable to celebrate the Mass.

Indeed, if the Pope himself has ordained a deaf priest, it's hard to argue that it's invalid to do so.

Various forms of sign language are considered true languagess

This argument declares that sign language is essentially a vernacular language. Fr. McNamara explains:

A second argument stems from the nature of sign language as a true language. Although obviously connected to the language of the country where the person lives, sign language is not just a paraphrase of the local spoken tongue but a true language with its own grammar, structure, and syntax. Many thousands of people use it as their ordinary means of communication and are capable of expressing the full range of human intercourse.

Assertions against giving the mass nonverbally appear to be exhortations against “verbum mentis”

Expanding on this point, Fr. McNamara writes:

It is true that traditional sacramental theology required vocalization for the sacramental form. This was one reason why deafness, along with many other physical impairments, was considered as an impediment to ordination. This would be the position found in most theological manuals. However, the closest we can come to an official pronouncement that vocalization is essential to validity is an assertion of Pope Pius XII who, with remarkable foresight, affirms the necessity for a concelebrant to speak the words of consecration. However, in this case, the Pope was asserting the invalidity of the “verbum mentis,” the simple mental or interior repetition of the sacramental form with no exterior sign. Thus a celebration by a priest who does not open his mouth but only reads interiorly the prayers would be invalid. This is not the case of the use of sign language which is not “verbum mentis.” In this case, the deaf priest is truly pronouncing the words of consecration, even though he is using visible and not audible words.

There is no official translation of the Latin Missal into sign language

Due to sign language being a relatively recent invention, and there not being a standardized way to write it down in print, an official translation has not been developed. However, he contrasts using unofficial translations with a priest paraphrasing prayers (something that is forbidden).

Even though the use of sign language is approved, there is still no official translation of the Latin Missal into sign language. Since sign language is such a recent phenomenon in the liturgy it will probably require some years to develop some of the more technical vocabulary and incorporate it into worship. A formal missal is probably even further away as there is not yet a fully standardized method for writing sign language, even if such a book is deemed necessary at all. Since sign language is a different language it cannot be compared to the situation of a priest who commits the abuse of paraphrasing the official prayers. A possible, albeit admittedly unsatisfactory, analogy for simultaneous speaking and signing would be the situation of observing a papal Mass on television in which the commentator reads the prayers from the English missal while the Holy Father is celebrating in Italian.

So how does the priest raise lift up the Eucharist while performing the mass in sign language?

In this video of an ASL mass, the priest holds up the Eucharist and says the words one-handed if able to.

When needing to hold both the Body and Blood, it appears that he says the the words before elevating it (you will notice that he changes the page of the liturgy book afterwards).

It seems like there is an alternative method if there is more than one priest. This mass (described in a news article here) involved the celebrant (i.e. primary priest) signing the words while the other priest elevated the Body and Blood.

I imagine that most other sign language masses are performed in this way, although given how these are relatively new, I imagine that this might be formally clarified in the future.