The balanced hypergraph partitioning problem (HGP) is to partition the vertex set of a hypergraph into k disjoint blocks of bounded weight, while minimizing an objective function defined on the hyperedges. Whereas real-world applications often use vertex and edge weights to accurately model the underlying problem, the HGP research community commonly works with unweighted instances. In this paper, we argue that, in the presence of vertex weights, current balance constraint definitions either yield infeasible partitioning problems or allow unnecessarily large imbalances and propose a new definition that overcomes these problems. We show that state-of-the-art hypergraph partitioners often struggle considerably with weighted instances and tight balance constraints (even with our new balance definition). Thus, we present a recursive-bipartitioning technique that is able to reliably compute balanced (and hence feasible) solutions. The proposed method balances the partition by pre-assigning a small subset of the heaviest vertices to the two blocks of each bipartition (using an algorithm originally developed for the job scheduling problem) and optimizes the actual partitioning objective on the remaining vertices. We integrate our algorithm into the multilevel hypergraph partitioner KaHyPar and show that our approach is able to compute balanced partitions of high quality on a diverse set of benchmark instances.