My boyfriend is not over his ex

I've been with my boyfriend for 6 months. Tomorrow we sign documents for a flat share together. In general, we treat each other well and do not fight often. We have a serious relationship and I have no doubts I love him. He told me he loved me on the 3rd month. Yesterday we had an honest talk, and he told me he hears from time to time with his ex girlfriend (they were friends since 2012 and dated for 2 years - 2013 and ended in 2015, lived together and remained friends). He decided to show me the chat and to be honest because for him it was not something important - they were exchanging 2-3 sentences in every 1.5 - 2 months. I saw something that broke my heart. On 15th August he wrote to her for her b-day and ended with " miss your face". He told me he loves me the first time on 27th August. I couldn't stop crying and couldn't sleep. For me he is not over her even after 2,5 years of separation. He installed this chat app that they use in China so he can remain in touch with his ex- Chinese. I told him how horrible I felt and that I can not accept him staying in touch like this otherwise I would not trust him because of that sentence. He told me it was over long time ago, he decided to delete the application, said he doesn't need that, and said I am the girl he can imagine himself being with the rest of his life (getting married, having kids, growing old with). He said I am the first girl he feels so serious with and he never imagined such a strong relationship before with anybody else. Do not know if it was a guilty talk or not. I think he is not over her. What shall I do about us?

I would just say be very very sure what you are doing . If flat is a rental it maybe ok and will give you time to think but do not buy

I would just say be very very sure what you are doing . If flat is a rental it maybe ok and will give you time to think but do not buy