: A novel crystalline structure of hybrid monolayer hexagonal boron nitride (BN) and graphene is predicted by means of the first-principles calculations. This material can be derived via boron or nitrogen atoms which are substituted by carbon atoms evenly in the graphitic BN with vacancies. The corresponding structure is constructed from a BN hexagonal ring linking an additional carbon atom. The unit cell is composed of seven atoms, three of which are boron atoms, three are nitrogen atoms, and one is a carbon atom. It shows a similar space structure as graphene, which is thus coined as g-B3N3C. Two stable topological types associated with the carbon bond formation, i.e., C-N or C-B bonds, are identified. Interestingly, distinct ground states of each type, depending on C-N or C-B bonds, and electronic bandgap as well as magnetic properties within this material have been studied systematically. Our work demonstrates a practical and efficient access to electronic properties of two-dimensional nanostructures, providing an approach to tackling open fundamental questions in bandgap-engineered devices and spintronics.