We form junctions between two single layers of p-type WSe2 and n-type MoS2 in both sequences. The WSe2|MoS2 and MoS2|WSe2 junctions of ultimate thickness limit exhibit current rectification when characterized vertically with a scanning tunneling microscope (STM) tip. The direction of rectification in the pn-junction is opposite to that of the np-junction, confirming occurrence of the rectification to be due to the junctions themselves. From scanning tunneling spectroscopy (STS) and correspondingly the density of states (DOS), we locate the conduction and valence band edges (CB and VB, respectively) of the materials inferring their single-layer and 2H phase configuration. Band edges of the semiconductors form a type-II band alignment resulting in current rectification. In junctions of WSe2 and MoS2 with single layers having a partial overlap, we map band edges along different points on individual semiconductors and the overlapped region (junction). The results have provided experimental evidence of current rectification through van der Waals vertical heterojunctions between two single layers.