Change the main menu for some loggedin users ?

Hey, quick question. I'm building a membership area on my new website, and I'm having a hard time figuring out the best site architecture for this membership area (because I'll have a lot of various content)...

My current idea is the following : when a member is logged in, replace the main Wordpress horizontal menu (let's call it "public-menu") by another Wordpress menu specifically dedicated to members (let's call it "member-menu")...

So basically, prospects would see the normal public menu, and customers, after they logged in, would see the membership menu in the same exact spot...

It would make sense as loggedin user won't need anything from the public menu, and vice-versa.

(see this demo page for exemple to see the menu if needed :

Now my problem is : I have no idea how I can make it so that non members see one menu on the "main menu location" (below the header, horizontal), and loggedin members see another menu at the same location ?

Any idea how to do that ?

Thanks a lot in advance,