Soloway is representing the couple and 11 others around the country who face deportation of a foreign-born spouse.
"At the time [DOMA] was passed, gay couples in the U.S. didn't have the opportunity to be married," he said. "Now, we finally have a critical mass of same-sex couples who are seeing the impact of DOMA."
Velandia immigrated to the U.S. in 2002 on a visitor's visa. He founded his own dance school, HotSalsaHot, and has appeared on the Spanish-language television show "Mira Quien Baila."
Vandiver and Velandia have lived together in New Jersey since 2006. Princeton recognized them as domestic partners in 2007, and this summer, they married in Connecticut, where same-sex marriage is legal.
"I started from zero in this country -- new language, new culture," said Velandia. "It's been like the American Dream."
But when his visa expired, an application for a green card was denied. In 2009, Velandia received a notice of deportation. If deported, he could be barred from entering the U.S. for 10 years.
According to the New York-based advocacy organization Immigration Equality, thousands of these gay couples -- one American and one an immigrant -- leave the U.S. each year for countries where gay immigrants are welcome.
Vandiver and Velandia say neither wants to leave the United States.
"My training is to be a professor and teach political theory in an American context," said Vandiver. "I really need to stay in America to pursue my studies and my work. I don't think I should have to leave America."
Velandia said he cannot imagine going back to Venezuela, where he lived a closeted life.
"With my relationship with Josh in America, I found the love of my life. I can actually be myself," he said. "It's ridiculous to try to go back to Venezuela being a gay man. I was repressed by the culture and religious beliefs. Going back would be like going into the past."
"We expect DOMA to be defeated in a few years and be history," said Soloway. "We want people like Henry to be here."
Carry Tucker, 55, of Sacramento, Calif., has been separated from her legal wife since they were married in Canada in 2007. Her spouse, Claire Pollard, 49, is a citizen of the United Kingdom.
"We tried every legal way to get Claire into the country to live with us," said Tucker, an Air Force veteran who calls herself a patriot. "We looked into everything."
The couple was prepared to move to Canada where Pollard had applied for legal residence under the skilled-worker program, but the economy went sour in 2008.
Under Gov. Arnold Schwarzenegger, California recognized legal same-sex marriages, but DOMA stood in their way.
They have now been separated for five years. Tucker says her 18-year-old daughter is a "stranger" to Pollard.
"All of her teen years, my daughter did not have her loving stepmother in her life," said Tucker. "They are the two loves of my life, and they don't know each other. That is the most painful part for me."
Same-sex marriage is legal in Connecticut, Iowa, Massachusetts, New Hampshire, Vermont and the District of Columbia, and is recognized by one Indian tribe in Oregon.
In California, same-sex marriages were performed for five months until voters enacted Proposition 8 in 2008. A federal judge has ruled the ban unconstitutional, and a court of appeals has issued a stay on the ruling, which is expected to work its way to the Supreme Court.