I have recent experience renewing a passport. I renewed my blue passport back in September and received the new one in about three weeks without paying the $60 fee. I received the old one back about a week later.
I know, here in DC, there are constraints on going to the passport office in terms of priorities and fees. Yes, you can generally get a passport that day if you are willing to wait literally almost all day for the largest fee. You can come back a day or two later for a lower fee.
In addition to the other advice, I would check with the nearest consulate or embassy of the country that issued you the visa. (I'm assuming it's a multi-entry visa?) Perhaps they can reissue you the visa in the new passport? Or, worst case, their immigration people may not consider a visa in an expired passport to be valid. I would do this first before doing anything with your current U.S. passport.