Dystopian YA MC is a teen age girl midwife

From what I remember of the story it is set in a place that has an inner city and an outer city. The outer city is the poor part and the inner city is the really rich part. Anyways the girl is the apprentice of the actual midwife and in the beginning of the book she's at a birth and aparently there is a rule about the babys because every third baby is taken to the inner ring to be raised there. So this womens baby is taken to the inner wall where it's taken from the MC and stuff happens, but another major part is there is a hanging and it's a pregnant women and her husband/boyfriend and after the women dies the MC does a C-section and saves the baby which was a big no no. So the MC flees the city and takes the baby with her and after that I don't remember much except there was nothing outside the city I think?
I read it about 2010-2014
Thanks for any help you can provide!

Birthmarked by Caragh M. O'Brien?

YES!! Thank you so much!!