What I do is, I leave the UV sphere where it is, but I place a mesh cylinder through the sphere (so they intersect), then I use the Knife tool and trace around where the cyclinder intersects.
Once I’ve done the whole circle, I just hit ‘P’ and delete the separate piece or delete vertices or whatever.
Because it’s a hand-drawn circle in effect, it’s not the most special solution on the planet, but the only way to do it perfectly supposedly is to use Booleans. For me, I try to avoid Booleans because they seem to stuff up your mesh (ick).
The method GreyBeard sounds OK too. However, I don’t think you will keep the integrity of the sphere if you do that.