The classical stochastic shortest path problem (SSPP) on integer-weighted Markov decision processes asks for a scheduler that maximizes the expected accumulated weight before reaching a goal state while ensuring that the goal is reached almost surely. In this article, we introduce the positively almost-surely terminating stochastic shortest path problem (PAST-SSPP), a variant of the classical SSPP, where the additional requirement is made that the expected number of steps for reaching the goal has to be finite. We show that PAST-SSPP can be solved in polynomial time. To this end, we provide an extension of the well-known spider construction as introduced by Christel Baier et al. used for the solution of the classical SSPP.