- Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. For several years, the best upper bound on the time to compute a GMDST was cubic with respect to the number of points in the input set. Recently, Timothy Chan introduced a subcubic time algorithm. In this paper we present an algorithm that generates a tree whose diameter is no more than (1+ ε) times that of a GMDST, for any ε> 0. Our algorithm reduces the problem to several grid-aligned versions of the problem and runs within time $ O (ε-3+ n) and space O (n).