“This is the most exciting thing that has ever happened to us,” the Today weekend anchor tells PEOPLE exclusively in its new issue. “But I don’t want to bring my daughter into a world where I’m not comfortable telling everyone who I am and who her mother is.”

“The beauty is that we live in a time where there’s no need for secrecy,” Gosk, 40, adds. “This is a spectacular moment for us.”

Together for three years, the pair began discussing starting a family and after deciding Wolfe would carry the child, underwent artificial insemination with an anonymous donor.

The case hinges on the fact that no doctors were used for the artificial insemination. The state argues that because William Marotta didn’t work through a clinic or doctor, as required by state law, he can be held responsible for about $6,000 that the child’s biological mother received through public assistance – as well as future child support.

Angela de Rocha, spokeswoman for the Kansas Department for Children and Families, said that when a single mother seeks benefits for a child, it’s routine for the department to try to determine the child’s paternity and require the father to make support payments to lessen the potential cost to taxpayers.

Marotta, a 46-year-old Topeka resident, answered an online ad in 2009 from a local couple, Angela Bauer and Jennifer Schreiner, who said they were seeking a sperm donor. After exchanging emails and meeting, the three signed an agreement relieving Marotta of any financial or paternal responsibility.

But instead of working with a doctor, Marotta agreed to drop off a container with his sperm at the couple’s home and the women successfully handled the artificial insemination themselves. Schreiner become pregnant with a girl.

Late last year, after she and Bauer broke up, Schreiner received public assistance from the state to help care for the girl.

The Kansas Department for Children and Families filed a court petition against Marotta in October, asking that he be required to reimburse the state for the benefits and make future child support payments. Marotta is asking that the case be dismissed, arguing that he’s not legally the child’s father, only a sperm donor.