Hosting Domain name

Hi,

I installed WordPress on a free hosting provider. Let’s say its address is http://www.hostingcompanyname.com/myblogname.

I also bought a domain name from another company. Let’s say the domain name is http://www.myblogname.com.

When I write http://www.myblogname.com on my browser, my blog shows up. Everything’s fine until here.

What happens is that when I put the mouse over any link, the browser reveals the hosting address, instead of showing the domain name. For example, if it’s a link to a page named Music, putting the mouse over it shows the address:
http://www.hostingcompanyname.com/myblogname/music, instead of
http://www.myblogname.com/music

I managed to change this by changing the Blog address (URL) in the General Options Administration Panel from http://www.hostingcompanyname.com/myblogname to http://www.myblogname.com, but then ALL the links on the blog STOP working: no matter what link I click at, it always takes me back to the blog’s root address (www.myblogname.com). The blog loses all the funcionality.
Even typing http://www.myblogname.com/music on the browser takes my back to the blogs root.

I appreciate for your support.