Biological Parents in IVF Baby Mix-Up Heartbroken Over Giving Up Custody

The biological parents in the Florida IVF mix-up are incredibly sad to give up custody of their daughter ... at least according to their lawyer.

Here's the deal ... Tiffany Score and Steven Mills gave birth to a baby, Shea, who isn't theirs after the Fertility Center of Orlando allegedly implanted the wrong embryo ... but they reached a custody agreement with the child's biological parents, allowing them to continue as the girl's permanent custodial parents.

Thing is ... the biological parents' attorney, Rob Marcereau, says his clients are devastated.

Marcereau told NBC News ... "They are heartbroken over what has happened, and they also understand that the birth couple are also suffering. They had to make the heartbreaking decision to not fight for custody."

He says the biological parents -- whose names have not been publicly announced -- would have faced an "incredibly uphill legal battle" if they wanted to seek custody of their daughter, because the laws in Florida and the U.S. favor the parents who actually give birth to the child.

Marcereau said the biological parents didn't feel a protracted legal battle would be in Shea's best interests ... and he said the custody agreement still allows them to be in the kid's life.

The lawyer says the decision wasn't easy ... and it came after several meetings with Tiffany and Steven, which he says were filled with "a lot of tears and hugs."

Marcereau said Shea's birth parents were "shocked" to learn their embryo was given to another couple.

He said it's "inexcusable" for the fertility clinic to make such a massive mistake, explaining ... "They have been placed in such a difficult situation that no one should be placed in, and the decision they have been forced to make has just been agonizing."