“Upon further investigation, we may be entitled to additional monetary damages if you have used other ancestry DNA programs, facial recognition tools on the internet or any other means, directly or indirectly, to contact or seek the identity of the donor,” said the letter, written by the sperm bank’s general counsel, Margaret Howell Benson.

“We will seek a restraining order or injunction if you continue with this course of action in any manner.”

NW Cryobank is known to be aggressive in enforcing its privacy provisions, Mr. Vorzimer added. “But now we’ve reached a point where it’s become impossible to protect the privacy of clients and sperm donors.”

Donor sperm is sold anonymously, but NW Cryobank and its affiliate California Cryobank — the largest sperm bank in the country — now only accept “ID disclosure” donors. The donor is required to agree that when a donor-conceived child turns 18, he or she is entitled to know the donor’s identity.

Leora Westbrook, general manager and vice president of NW Cryobank, said in an email that the bank does not prohibit clients or their offspring from taking a DNA test. But “we seek to prevent the use of that information to identify a donor who has made a donation in reliance upon anonymity.”

Once the child is no longer a minor, Ms. Westbrook added, he or she may not only take the DNA test but may also contact the bank to determine if the donor is open to being contacted.