Gay married man who's undocumented immigrant fights deportation in California March 22, 2012 -- Updated 1953 GMT (0353 HKT) Alfonso Garcia, 35, undocumented immigrant from Mexico, faces deportation He and Brian Willingham, 37, were married in New York The couple says Garcia should be allowed to stay here as the husband of U.S. citizen Defense of Marriage Act defines marriage as between opposite sex couples Los Angeles (CNN) -- A gay married California couple was scheduled to appear Thursday in federal immigration court in San Francisco in an effort to halt the deportation to Mexico of one partner who's an undocumented immigrant, their attorney said. Alfonso Garcia, 35, who came to the United States as a boy with his parents, will ask a federal immigration judge to put his deportation proceeding on hold as he and husband Brian Willingham, 37, petition the federal immigration service for legal residency based on their marriage, said attorney Lavi Soloway. The couple lawfully married in New York and are now registered domestic partners in California living in the San Francisco Bay area, but the federal immigration court doesn't recognize gay marriage under the federal Defense of Marriage Act, which defines marriage as between opposite sex couples, Soloway said. The federal law is now being challenged on constitutional grounds, with rulings expected this summer in federal appeals court, but the case hasn't yet reached the U.S. Supreme Court, Soloway said. "If they were an opposite sex couple, we wouldn't have this discussion right now," Soloway said about Garcia's efforts to secure legal U.S. residency as a man married to a U.S. citizen. Willingham is a U.S. citizen. "What this case is about is a Mexican man who was brought to the United States as a child and has lived here for 20 years, as has his whole family," Soloway said. But he doesn't have lawful status. "We have a whole campaign around this case and other cases like it," Soloway said, referring to the Stop the Deportations campaign and its website, in which gay and lesbian binational couples are fighting deportation, separation and exile caused by the Defense of Marriage Act and U.S. immigration law. Garcia's undocumented status was learned by authorities during a routine traffic stop last July, which led to a background check, the couple said in a statement. Garcia's parents are legal residents applying for U.S. citizenship, the attorney said. Garcia and Willingham met in October 2001. "As a gay American citizen the federal government offers me zero, zilch, nada, null access to the federal rights that all married couples have," Willingham said on the couple's Web page. This is not an issue of separate, but equal. There are no separate federal rights for married gay couples. There are no rights at all. This is not a front of the bus, back of the bus issue. This is the federal government telling us to get the hell off of the bus. President Barack Obama has called for a legislative repeal of the Defense of Marriage Act, and while Democrats on the Senate Judiciary last year voted to send a repeal to the full Senate floor, the measure is considered as having no chance of getting passed by the Republican-led House. If Garcia is deported, he would be barred from returned to the United States for 10 years, his attorney said. I've spent most of my life in the United States. This country is my home, and Brian is my husband. I don't want to lose everything we have built together and be told I can't come back to the U.S. for 10 years. I just want to know we can be together. I just want to know the solemn oath we made to one another will count in the eyes of the law," Garcia said in a statement.