Domain mapping issue

I have a blog on WordPress.com. I bought a new domain name through Gandi.net and mapped my new domain name to my WordPress blog – pointing the DNS servers to the wordpress.com servers.

I’ve noticed that when I type my new domain in the address bar, it directs it to my blog but then the domain changes to the old wordpress domain name. How can I make sure that the new domain name stays in the address bar when people visit my WordPress site? What steps would I need to take to make this happen?

Gandi support said I will need to check with wordpress on how to do this as they don’t have access to the records.

Thank you for your help!
Heather