What is a good way to show your best friend you care about the friendship?

My best friend has chose to end our friendship over pretty much a misunderstanding of something that happened between us. He feels like I betrayed his trust and in a way I did. He thinks that I had been faking our friendship and had alterior motives to why I wanted to be friends. He is so wrong about it and his friendship means the world. He's become like a brother to me and I really want to prove that his friendship matters. What can I do just to show him that?