M.C.
It looks like you might need a clariication of terms here. FMLA covers THREE SEPARATE situations, two of which usually overlap in the case of a woman giving birth:
1) your own medical inability to work
2) caring for a newborn or newly adopted child
3) caring for an immediate family member who is ill and in need of constant care.
Under most situations, your employer is required to give you 12 weeks of UNPAID leave per year to cover any combination of these situation. The 12 weeks is cumulative over the year--so, for instance, if you take 8 weeks to care for your child when she's born, then you get injured 6 months later and are out for 4 weeks: that's your 12.
Now, for women who give birth, obviously, 1 and 2 usually overlap. In addition, many employers allow you to take PAID sick or disability leave towards those 12 weeks--so, for instance, you might go on 6 weeks sick leave after your child is born, and still get paid. It is up to your employer whether this paid leave counts towards your 12 wks FMLA or not. So, for instance, when my DD was born, I was entitled to 8 wks sick leave, since I had a C-section. The way my employer's policy was written, I was then legally entitled to an additional 12 weeks unpaid under FMLA (although I couldn't afford to take it). If your employer only offers unpaid leave, however, that gets rid of at least this complication.
So what you're looking at before the baby is born is that, as others have noted, if your doctor gives you a note, you may qualify for FMLA under situtation (1) above. Immediately after the child is born, you will likely qualify for (1) and (2) at the same time. Then, once you feel better, if you still haven't used your 12 weeks, you'll still qualify for (2).
And, incidentally, (2) works for men, too. Your husband is legally entitled to take 12 weeks unpaid leave to care for your newborn anytime within one year of the child's birth. Most men don't use it--and my husband's boss laughed when DH brought it up--but, legally, it is true.
Most people--including HR folks--unfortunately don't understand FMLA. Admittedly, it is complicated, but it's worth figuring out to insist upon your rights. I happen to be a professor, and got most of my info by reading two booklets published by my professional organization about the ways in which FMLA applies to my professional situation: "The Family and Medical Leave Act: questions and answers" and "Pregnancy and the Academy: questions and answers," both published by the American Academy of University Professors. Both these books were written by lawyers, but in an accessible style so non-lawyers can understand. You could probably find similar books at your local library, if you need some ammunition to bring with you when you talk to your employer.
Good luck with everything, and congratulations on your baby girl!