Two-dimensional boron sheets (that is, borophene) have recently been realized experimentally and found to have promising electronic properties. Because electronic devices and systems require the integration of multiple materials with well-defined interfaces, it is of high interest to identify chemical methods for forming atomically abrupt heterostructures between borophene and electronically distinct materials. Toward this end, we demonstrate the self-assembly of lateral heterostructures between borophene and perylene-3,4,9,10-tetracarboxylic dianhydride (PTCDA). These lateral heterostructures spontaneously form upon deposition of PTCDA onto submonolayer borophene on Ag(111) substrates as a result of the higher adsorption enthalpy of PTCDA on Ag(111) and lateral hydrogen bonding among PTCDA molecules, as demonstrated by molecular dynamics simulations. In situ x-ray photoelectron spectroscopy confirms the weak chemical interaction between borophene and PTCDA, while molecular-resolution ultrahigh-vacuum scanning tunneling microscopy and spectroscopy reveal an electronically abrupt interface at the borophene/PTCDA lateral heterostructure interface. As the first demonstration of a borophene-based heterostructure, this work will inform emerging efforts to integrate borophene into nanoelectronic applications.