Sara Hart Weir, the president of the National Down Syndrome Society, said that the separation of the girl from her mother was “a very serious matter” and that “we expect all individuals to work together and find solutions and put any political rhetoric aside to help this young girl.”

“Our hearts and prayers go out to this child and her family during this unprecedented and trying time,” Ms. Weir said Wednesday morning. “N.D.S.S. will work with the appropriate agencies to ensure she receives all the resources from us that she needs to help comfort her until she can be reunited with her father and eventually her whole family, where she belongs.”

Customs and Border Protection said that it came into contact with the girl, four of her siblings and her mother on June 3, when they were stopped in a car. The driver of the car was a United States citizen who was trying to smuggle the family into the country, the department said.

Three of the children, who are United States citizens, were released to an aunt. The 10-year-old girl and a sibling were placed into the custody of the Department of Health and Human Services.

While the mother told the authorities she was in the country illegally, Customs and Border Protection said she had not been charged with a crime.

“The mother was not prosecuted, but is instead being held as a material witness to support the prosecution of the smuggler, which precipitated the separation of the two other children, both Mexican citizens,” the department said in a statement. “This smuggler has a criminal history including a flight, escape, aiding and abetting making it important that we prosecute.”

The foreign minister of Mexico, Luis Videgaray, discussed the girl’s situation at a news conference in Mexico City on Tuesday and called the Trump administration’s separation policy “cruel and inhumane.” He said that the girl and her brother were sent to a facility in McAllen, Tex., while their mother was moved to Brownsville, about 60 miles away.