Summary: A method which rapidly computes optimum finite-thrust space trajectories is presented. The approach is based on the approximation of the state time history by a polynomial. Differentiation and integration are performed in closed form and the problem is reduced to one of ordinary calculus. The technique is applied to constant power, variable-thrust trajectories, and results are presented for Earth-Mars transfers. Apparently the method is an order of magnitude faster than standard techniques. The method is not restricted to low-thrust levels or to transfers between nearby orbits.