The chemical functionalization of graphene in ultra-high vacuum (UHV) with monolayers of 3,4,9,10-perylene-tetracarboxylic dianhydride (PTCDA) has been characterized with scanning tunneling microscopy (STM). The molecules self-assemble into stable, well-ordered monolayers that are arranged in a herringbone phase with extended domains spanning hundreds of nanometers. At submonolayer coverage, PTCDA forms stable, isolated molecular islands. The molecular ordering is unperturbed by either defects in the epitaxial graphene or atomic steps in the underlying SiC surface. Scanning tunneling spectroscopy (STS) performed on the PTCDA monolayers reveals strong features in the electronic density of states that are distinct from the pristine graphene regions. The demonstration of robust, uniform organic functionalization of epitaxial graphene is an important step toward realizing graphene-based molecular electronic and sensing devices.