In the Steiner Tree problem, we are given as input a connected n-vertex graph with edge weights in \{ 1, 2, . . ., W\} , and a set of k terminal vertices. Our task is to compute a minimum-weight tree that contains all of the terminals. The main result of the paper is an algorithm solving Steiner Tree in time O (7.97 k · n 4 · log W) and using O (n 3 · log nW · log k) space. This is the first single-exponential time, polynomial space FPT algorithm for the weighted Steiner Tree problem. Whereas our main result seeks to optimize the polynomial dependency in n for both the running time and space usage, it is possible to trade between polynomial dependence in n and the single-exponential dependence in k to obtain faster running time as a function of k, but at the cost of increased running time and space usage as a function of n. In particular, we show that there exists a polynomial space algorithm for Steiner Tree running in O (6.751 k n O (1) log W) time. Finally, by pushing such a trade-off between a polynomial in n and an exponential in k dependencies, we show that for any ε > 0 there is an n O (f(ε )) log W space 4 (1+ ε ) k n O (f(ε )) log W time algorithm for Steiner Tree, where f is a computable function depending only on ε . © 2019 SIAM.