We consider the problem of determining whether a t-sparse or lacunary polynomial f is a perfect power, that is, f=h^r for some other polynomial h and integer r, and of finding h and r should they exist. We show how to determine if f is a perfect power in time polynomial in t and log(deg(f)), i.e., polynomial in the size of the lacunary representation. The algorithm works over GF(q)[x] (at least for large characteristic) and over Z[x], where the cost is also polynomial in log(||f||). Subject to a conjecture, we show how to find h if it exists via a kind of sparse Newton iteration, again in time polynomial in the size of the sparse representation. Finally, we demonstrate an implementation using the C++ library NTL.