I'm in the same exact situation, and it SUCKS. I need to make a payment of almost $400 by the end of May and I have no idea how I'm going to get that done, let alone June's payment. We have so much on our plate right now, and in addition to the payments, I need to save so that we have money to live on while I'm on maternity leave because I don't get paid leave.
Do you have good credit? I ask because you can apply for Care Credit and use that to pay for medical expenses. I don't use credit cards but that may be an option for you.
Another option would be to find out if you qualify for Medicaid. My regular OB accepts Medicaid so if I did qualify (which I don't, I make too much) I could still see the same doctor, which is nice (but sort of irks me because there are women there getting free care for popping out babies they can't afford when I have to pay, but that's a topic for another day... LOL).
I haven't heard of any OB offices that do not have this procedure in place. They have to insure that they will get paid in some way. Some of them draw out their payment plans longer, though. My OB requires all payments before week 24 of pregnancy, but I have heard of other patients negotiating their payments stretched out over the duration of their pregnancy, making smaller payments. The OB I had for my first daughter required all payments to be paid by week 28.
Good luck! Pregnancy is a wonderful time, I wish you didn't have to stress about money!