You can set up URL forwarding from who ever you registered your domain name with, or you could set up a 301 redirect on the temporary domain name, this will probably be a good idea for SEO reasons, though i'm no expert!

Yup, two ways. Either use URL forwarding through the domain registrar, or if you already have nameservers set up with a webhost on the second site, just use a URL redirect through a meta refresh tag in the html head. Second method is preferred, 'cause then your website will be live and all ready to go with a simple swap of index.htm files. Plus it can be spidered with the description information and start the site's population on search engines.

<meta http-equiv="refresh" content="0;url=http://othersite.com/" />

The number after content is the number of seconds to wait before refreshing. I usually leave a hotlink in the body copy too, saying something like "This page should automatically redirect you. If it doesn't, please click HERE".

Use a redirect file to point people from the one site to the other.

Here is the one I use:

Replace yourURLhere with the URL you want people to go to. Then make this html page the only one on the site you want to redirect from.

Thanks for the options everyone. I won't be able to do the URL redirect because the hosting currently attached to the AltSite.com domain is going to get canceled.

It looks like URL forwarding costs money through my domain name registrar, there isn't any other way to get that domain to point to another site/host?

Wha-!? Get a new registrar. I've never heard of anything so ridiculous as charging for domain forwarding, and I've been hosting websites for many years and have used many domain registries.