Good Burger by the DC Zoo
If you really mean "near the zoo" then of course Palena.
Dupont Circle - BGR is OK, not great.
Considering that the walk from the zoo to Palena is not bad, or you could take a taxi for a couple of bucks, or even walk down the hill on Connecticut to Woodley Park Metro and then up one stop to Cleveland Park (if walking uphill is a problem for you)...
or if you really did mean DuPont Circle, then two metro stops...
You could do Palena easily from either starting point.
3529 Connecticut Ave NW, Washington, DC 20008
Near the zoo or near Dupont Circle? I'll go with the latter and recommend BGR. I think it falls somewhat sort of "amazing" but I think it's very good and the best in that area. The last time I had a shake there it was very disappointing. Kind of icy, like the ice cream had been stored at the wrong temperature or something.