Can my child have my last name is the mother is still legally married to someone else?

I recently had a child with a woman that was separated but "legally" still married to someone else. Not only that but I had suspicion of her sleeping around on me before she got pregnant. We are no longer together and now the child is 3 weeks old. Because of her actions I wanted to get a paternity test to be 100% positive he was mine. I have now taken the test and will receive the results withing a week. He currently has her Maiden name as his last name on the birth certificate and she says because she is still legally married I can't change his last name to mine, even with paternity test results. Is this true? If not, how to I go about the process of doing so. If he's mine I clearly want him to have my last name and to have a relationship with him. Any suggestions or advice would be helpful. Thanks