You could use the newspaper and mulch route, maybe use cardboard instead of newspaper so it lasts longer, then a light layer of mulch. You just don't want to bury the existing soil surface a foot deep with new soil, but a light layer of mulch won't hurt.

You could also lay down landscaping fabric and mulch. As long as there was no green growth from the tree, and the roots are not exposed on the surface, you could use glyphosate to take out the weeds.

Or, you could just run the mower over the weeds to keep them under control and let them be part of your lawn.

The simplest, easiest, and least expensive way to control "weeds" growing around and under the drip line of any tree is with newspaper covered with a mulch. As long as the mulch layer is not terribly thick there is no concern about "killing" the tree by smothering the roots.
Last time I looked at one of the glyphosate products it was so expensive that I simply shuddered and walked away.

Thanks beeone and kimmsr. I'll try the newspaper/mulch route. A few roots are exposed so don't want to use glyphosate.

Thanks.