We have devised a DNA encoding method and a corresponding DNA algorithm for the minimum spanning tree problem, an instance of optimization problems on weighted graphs. In order to find out the minimum spanning trees of a weighted graph G= (V,E,W) by means of molecular biology techniques, we encode each vertex vi\inV using one recognition code of length l, l=max{log_4n,6}; we encode each edge e_ij\inE using two DNA strands of length 2p=2max{w_ij, l}; for any two adjacent edges e_ij,e_jk, we add one DNA strand s_aijk of length w_ij+w_jk as an additional code. We also presented a DNA algorithm for the minimum spanning tree problem based on the proposed DNA encoding method, in which we firstly obtain the Euler cycle corresponding to the minimum spanning tree by means of the molecular biology techniques, and then the Euler cycle is converted to the minimum spanning tree. Our work provides further evidence for the ability of DNA computing to solve numerical optimization problems.