Extruding beveled edges on a cube

Starting with a cube, I want to transform the top and bottom faces to be a certain dimension (3x2, let’s say).
I want the other four sides of the cube to extend outward from the transformed faces and be beveled. I want the beveled top and bottom edges to touch so that a cross section through them would reveal a semicircle (or that a cross section through the entire shape would look like a circle that was extruded along the x axis). However, I don’t want the bevel to affect the original 3x2 top or bottom face at all. The bevel should start external to the original 3x2 dimensions of the top and bottom faces.

The effect I’m going for would have edges something like this:


but with the 3x2 dimensions of the top and bottom faces completely preserved.

I have tried several things–beveling edges and/or faces of the starting cube, pulling the lateral faces out from the cube and then trying to bevel those, etc. and I got the effect once before I realized what I was doing and now cannot seem to reproduce it. I have watched bevel and manipulation tutorial videos and searched the forums here and am out of ideas. I would appreciate a walkthrough of the steps aimed at a new user.