What's the easiest method to integrate a legacy phpBB forum right into a new Joomla site?

Essentially I wish to keep your navigation links (top and primary menu), and embed the forum as pages within the site.

Is achievable?

If you want user integration for Joomla and phpBB you should use JFusion, so people drenched in Joomla is going to be drenched in phpBB. Registered customers in Joomla is going to be duplicated to phpBB too (and the other way around, you need to setup JFusion configuration to determine which product is the "master"). I have used JFusion having a Moodle and Joomla installation and it is working pretty much.

To "integrate" your forum for your Joomla installation, you've got a fast implementation: inside your top menu, produce a link and alter it's Menu Item Type to Wrapper. In Parameters, set Wrapper URL for your phpBB path. By doing this you'll have an iframe together with your forum.

Should you prefer a better quality solution, I'd attempt to create a component.

I'd theme the forum properly and use it a sub-domain like http://forum.mydomain.com. By doing this it's unlikely to clash using the relaxation of the website.