Decomposing hypergraphs into simple hypertrees

Let T be a simple k-uniform hypertree with t edges. It is shown that if H is any k-uniform hypergraph with n vertices and with minimum degree at least n^k-1/2^k-1 (k-1)! (1 + o(1)), and the number of edges of H is a multiple of t then H has a T-decomposition. This result is asymptotically best possible for all simple hypertrees with at least two edges.