Air carrier with ZTL deviating for weather and issued a climb from FL320 to FL340 received TCAS RA.

Narrative

While climbing to our assigned altitude of FL320 with ATL Center; we asked if FL340 was available. ATL Center then cleared us to FL340. We asked for a deviation south of course around weather and were then switched to MEM Center. At approximately FL325; we saw a TA at FL330 and queried ATC. At exactly the same time he responded; we received an RA and immediately disconnected the autopilot and smoothly initiated a commanded 2000 FPM descent. Clear of conflict was announced by the TCAS as the aircraft passed overhead. The controller gave us a turn but by then we were already responding and following the guidance of the TCAS system. ATC never issued a traffic advisory to us or the other aircraft. We responded to a TA; queried ATC then responded and reacted to a RA. The ATL Controller evidently didn't enter our amended clearance altitude in to the ATC computer. When we checked in with MEM Center; he didn't verify our assigned altitude that we were cleared to maintain.Although ours was fully operational; NEVER NEVER NEVER DISPATCH AN AIRCRAFT WITH THE TCAS INOPERATIVE. Did I say NEVER?? I thought I said NEVER. Yep; I did. The TCAS saved the lives of all of us onboard.