I think all you need to do is back up the actual files, and the database (unless the new location is on the same machine).. Then just edit /blogroot/conf/_basic_config.php and look for the following sections:

Edit this section if you're moving the entire thing to a different host, so the db name and username will likely be different:

$db_config = array(
	'user'     => 'DBUSERNAME',     // your MySQL username
	'password' => 'DBPASSWORD',     // ...and password
	'name'     => 'DBNAME',  // the name of the database
	'host'     => 'localhost',    // MySQL Server (typically 'localhost')
);

Then scroll farther down and update this section to reflect the location/address of your blog:

/**
 * $baseurl is where your blogs reside by default. CHECK THIS CAREFULLY or nothing will work.
 * It should be set to the URL where you can find the blog templates and/or the blog stub files,
 * that means index.php, blog1.php, blog2.php, etc. as well as admin.php.
 * Note: Blogs can be in subdirectories of the baseurl. However, no blog should be outside
 * of there, or some tricky things may fail (including intempestive logouts)
 *
 * IMPORTANT: If you want to test b2evolution on your local machine, do NOT use that machine's
 * name in the $baseurl!
 * For example, if your machine is called HOMER, do not use http://homer/b2evolution/blogs/ !
 * Use http://localhost/b2evolution/blogs/ instead. And log in on localhost too, not homer!
 * If you don't, login cookies will not hold.
 *
 * @global string $baseurl
 */
$baseurl = 'http://www.yourdomain.com/path/to/blog/';

jj.

@phatchopolis the process is exactly as described above.

but want a back up of the current site for browsing

I assume that you want two sites, lets say, one for production (the new domain) and another as an archive (your current domain). Basically, as @jibberjab said, you need to make a backup of all the files and the database, and upload them to the new domain. Tip: some hosting companies offer this kind of services to their customer, all you need to do is tell the support staff what you need, maybe something like this: "I need to make an exact copy of the files from 'domainA.com' to 'domainB.com', also from the database A to database B". So all you need to do is to change the $baseurl at the /blogroot/conf/_basic_config.php in the new site.

Just as a note, if you change the domain, all the indexed links to your current blog will be lost. For example: if one of your posts appears in google's results, that link will be broken. You should consider to make 301 redirections to manage those kind of situations.

Regards!