Consider a directed graph G=V,E) with n vertices and a root vertex r∈V. The DMDST problem for G is one of constructing a spanning tree rooted at r, whose maximal degree is the smallest among all such spanning trees. The problem is known to be NP-hard. A quasipolynomial time approximation algorithm for this problem is presented. The algorithm finds a spanning tree whose maximal degree is at most O(Δ^*) + log n) where, Δ^* is the degree of some optimal tree for the problem. The running time of the algorithm is shown to be O n log n ^{log n}. Experimental results are presented showing that the actual running time of the algorithm is much smaller in practice.