Can my ex get full custody of our child if my current gf is a convicted felon?

I just recently found out that the girl I've been dating for about 5 months now is a convicted felon. She's been arrested several times in the past but I would never have know this if I hadn't have done a background check on her myself. She had told me that she had been arrested before but didn't say why. I can't specifically remember her charges but I know none of them involved anything that had to do with being dangerous towards children. We'll my ex( child's mother) recently found all of this out for herself and I an afraid she will take me for full custody. My gf is very good with my daughter, has never posed a threat to her in any way and it's been at least 4 years since her last arrest. He is a totally different person now and my ex has never had a problem with my gf watching our daughter before so I don't see that her finding this out should change anything. Please help me.