Nate, 29, captured the one-in-80,000 moment on camera moments after Annabelle was born weighing 3.7 kilos on February 21.
He and Rachael, 23, from Buffalo, New York, said they are celebrating Annabelle's healthy arrival - and the fact she was born with a caul is the ‘cherry on top’.
‘Everything happened so fast,’ Nate, a digital optimisation specialist, tells.
Scroll down to see the pics
‘Rachael couldn't see what was going on because she was behind a curtain and a little drugged up, but I was on the other side watching the operation.
‘As they started pulling things out of her stomach this bubble comes out and at first I thought it was part of her insides.
‘I knew it was weird because I had watched the C-section for my son 17 months earlier and I didn't see that.
Scroll down for the pics
‘All of the nurses that were around helping the doctor were getting excited and pointing at it, but I was confused and I didn't know what I was looking at.
‘I got a little scared because I couldn't tell if the nurses were reacting to something bad or good, so I looked over and was like, 'Oh my God, is everything OK?'
"The nurse didn't even answer she just said, 'Quick, take a picture,' so I pulled out the camera and snapped a few photos.
‘After that I went back with Rachael. I could tell she was out of it and wanted to reassure her.
‘There was a lot of pushing and tugging and the she was out, just like that. I'm not sure if they sliced it open or pulled it over her.
‘I'm so happy that I got the moment on camera. We are so happy to have her with her now and that was just the cherry on top.’
Rachael shared the photos on Instagram @astoldbymama, where she documents her journey through motherhood with Annabelle and her 17-month-old son Isaiah.
The blogger and full-time mum said: ‘My doctor said this is super rare and it was definitely a surprise for us.
‘I was taken aback when I saw the photos. It is completely amazing.’
The caul is a thin, filmy membrane, the amnion, that can cover a newborn's head and face immediately after birth, according to howstuffworks.
Babies can also be born 'en caul', where they emerge fully encased within the amniotic sac, which is filled with fluid in which the unborn baby floats and moves.
This fluid protects against infection and cushions the baby from bumps and injury as well as providing them with fluids that they can breathe and swallow.
Superstition surrounds the rare phenomenon of en caul births, with some cultures believing it is a mark of good luck.
To follow Annabelle's journey, click here