Alternating-time Temporal Logic (ATL) is a branching-time temporal logic that naturally describes computations of multi-agent distributed systems and multi-player games. In particular, ATL explicitly allows for the expression of coalitional ability in such situations. We present an automata-based decision procedure for ATL, by translating the satisfiability problem for ATL to the nonemptiness problem for alternating automata on infinite trees. The key result that enables this translation is a bounded-branching tree model theorem for ATL, proving that a satisfiable formula is also satisfiable in a tree model of bounded branching degree. It follows that ATL is decidable in exponential time, which is also the optimal complexity: satisfiability in CTL, a fragment of ATL, is EXPTIME-complete. The paper thus answers a fundamental logical question about ATL and provides an example of how alternation in automata may elegantly express game-like transitions.