2 Domains 1 IP Address

[markdamo]

We have recently bought a new domain www.example.com and are trying to get customer etc to associate us with this url intead of the old address - www.My-exampl.com.

Both URLs are stored pointing to the same IP address but I wanted to know if there is a way to influence Google to index the new address and remove old address so as people who search for us will go to the new url address instead of the old.