Larry Birkhead has finished the process of having his infant daughter's name officially changed on her Bahamian birth certificate.
The girl's name was previously Dannielynn Hope Smith.
The judge reportedly said he was "a little perplexed" as to "why this order is in front of us," adding he felt it was "weak" before denying the appeal.
Arthur also also ordered to pay court costs of $3,000, including Birkhead's fees.
This was an "expensive day for her," said Birkhead afterward. "I feel like I just got a divorce from her, and I didn't even know we had a baby together."
He planned to take Dannielynn back to his home in Louisville.