In this paper, we investigate the partial inverse knapsack problem, a bilevel optimization problem in which the follower solves a classical 0/1-knapsack problem with item profit values comprised of a fixed part and a modification determined by the leader. Specifically, the leader problem seeks a minimal change to given item profits such that there is a follower optimum selecting or discarding items according to two given disjoint subsets. Based on the bilinear value-function reformulation, we develop a single-level linear mixed-integer model for the problem and propose a branch-and-cut solution algorithm. Our computational experiments demonstrate the overall viability of our approach and that utilizing a new exponential-size class of valid inequalities for the leader allows to substantially reduce the overall solution times.