Should we break up over him talking to his ex girlfriend?

Just a few pieces of information before I delve any further into my question. I have been with my boyfriend for 3 months, whereas he was in a relationship with his last girlfriend for over 2 years. When we met they'd been broken up a few months and he was still hurting from it. When he eventually asked me to be his girlfriend, he told me it was because he was finally over her and he wanted to be with me, which I was happy to hear.

He has been bitter about her for a while, but somehow he claims they still maintain a friendship. It didn't bother me before that they ocassionally said hi and were pleasant to eachother, but today I discovered he was talking to her online at the same time as me, and she was getting most of his attention. Maybe it's just jealousy, which I'm sure makes up a huge part of it, but I feel like if it hurts me too much to have him still be friends with her and put her before me at times, he shouldn't be talking to her anymore.

He basically says she has been too big a part of his life for too long and he can't just get rid of her. I see that as him still having feelings for her, and if he really was in love with me then he would see how this is a problem for me and instead of being worried about hurting her by saying goodbye he'd want to stop hurting me by continuing the friendship.

Currently, he is 'thinking things through' over whether or not to stop talking to her, or break up with me. The fact it's even a choice hurts.