The emergence of exotic quantum phenomena is often triggered by a subtle change in the crystal phase. Transition metal dichalcogenides (TMDs) exhibit a wide variety of novel properties, depending on their crystal phases, which can be trigonal prismatic (2H) or octahedral (1T). Bulk NbSe2 crystallizes into the 2H phase, and the charge density wave and the superconductivity emerge simultaneously and interact with each other, thereby creating various anomalous properties. However, these properties and their interplay in another polymorph, 1T-NbSe2, have remained unclear because of the difficulty of synthesizing it. Here we report the first experimental realization of a monolayer 1T-NbSe2 crystal grown epitaxially on bilayer graphene. In contrast with 2H-NbSe2, monolayer 1T-NbSe2 was found to be a Mott insulator, with an energy gap of 0.4 eV. We also found that the insulating 1T and metallic 2H phases can be selectively fabricated by simply controlling the substrate temperature during epitaxy. The present results open a path to crystal-phase engineering based on TMDs.