We report a van der Waals heterostructure formed by monolayers of MoS2 and ReS2 with a type-I band alignment. First-principle calculations show that in this heterostructure, both the conduction band minimum and the valence band maximum are located in the ReS2 layer. This configuration is different from previously accomplished type-II van der Waals heterostructures where electrons and holes reside in different layers. The type-I nature of this heterostructure is evident by photocarrier dynamics observed by transient absorption measurements. We found that carriers injected in MoS2 transfer to ReS2 in about 1 ps, while no charge transfer was observed when carriers are injected in ReS2. The carrier lifetime in the heterostructure is similar to that in monolayer ReS2, further confirming the lack of charge separation. We attribute the slower transfer time to the incoherent nature of the charge transfer due to the different crystal structures of the two materials forming the heterostructure. The demonstrated type-I semiconducting van der Waals heterostructure provides new ways to utilize two-dimensional materials for light emission applications, and a new platform to study light-matter interaction in atomically thin materials with strong confinement of electrons and holes.