Rel Canonical issues for two urls sharing same IP address

[GravitateOnline]

Our client built a wordpress site on url A, then opted for a better url B. Rather than moving all the wordpress files/website over to the new url B, they just contacted GoDaddy, who hosted BOTH urls under the same IP address.

When I do a term target on url B, I'm flagged for rel canonical use. I can only get a B grade for each keyword. (I've also tried using url A, but I get the same flag and B grade results).

I'm not sure if this set-up will thwart our seo efforts for the site, because only the homepage comes up when you type in url B anyway. Every subsequent page displays the original url A. Somewhere, wordpress is also adding a rel canonical link on the homepage source to url A, too, which we can't seem to edit.

So, question is: is it ok to leave this set up as is with both urls hosted on the same IP address, or should we move the whole site over to the desired url B?

Thanks much!

[GravitateOnline]

Thanks for your answer, John. So, we should have the client 301 the url A to the desired url B. Confirmed!

[john4math]

This doesn't sound ideal.  So only the home page is under URL B, and the rest of the pages are hosted under URL A?  That would seem really strange to me as a user if I entered through URL B, went to another page, and the domain changed completely.

Ideally, you should 301 redirect everything under URL A to URL B rather than using rel=canonical for them.  There's no reason to host two identical sites like this.  It's fine for multiple sites to be hosted under the same IP.  Here's a really good SEOMoz blog post about using 301 redirects vs.canonical tags.