Now, she's fighting to keep the eight-week-old babies and the court decision could be a landmark case.
The biological father of the twins says she was merely a surrogate. But she says they had a deal to share custody and raise the kids together.
A lot of people in the legal community say this is really a first and it could broaden or narrow the definition of what a mother is and whether biological connection to a child has anything to do with it.
"This is my lifelong dream. I've always wanted to be a mom," Cindy Close said.
Close is at the center of what could be a landmark case in motherhood.
"I love them. They're wonderful, and I wish I could spend more time with them," she said.
On July 27, she gave birth to a boy and a girl.
"My son is much quieter than my daughter. She's a little diva. She's wonderful," Close said.
But Marvin McMurrey, the man who fathered them, says Close is not their mother and he's filed suit asking the court to agree with him and strip her of any rights to see them.
"I hope that we can settle this in the courts, and that I'll be able to bring my children home, and raise them with his help. That was always the plan," Close said.
That plan was hatched five years ago, when Close says she and McMurrey agreed to have children together despite no romantic relationship. But they learned she couldn't have children and so years later, with his sperm and anonymous eggs, she was impregnated.
Close says in court documents it wasn't until the day the children were born that she learned McMurrey was gay, was planning to raise the children with his partner and that she was just a surrogate.
"He's known for years that my only dream in life was to be a mom," Close said.
She says there was never a surrogate agreement in writing and despite no genetic link to the kids, they are hers.
"The law is, in our opinion, clear. Because she gave birth, she is therefore the mother of her two children," Close's attorney, Grady Reiff, said.
McMurrey's attorney declined to discuss the case with us but an independent attorney tells us she thinks Close is legally the mother.
"Under Texas law, when a woman gives birth she is either surrogate or mom. And in this case, it seems like she would be mom," attorney Sarah Frazier said.
Incredibly, if Close wins she still wants McMurrey to be in the children's lives.
"He is their father, just like I am their mother," she said.
On Monday, both sides are expected to appear in court to ask a judge to decide whether Close is actually the mother by law. If she is, then this will become a custody case.