One-dimensional graphene nanoribbons have recently attracted much interest as they have been recognized as promising building blocks for nanoelectronic devices. A mono-layer of graphene does not have a bandgap and is thus not appropriate for electronic applications. However, by patterning a graphene sheet into very narrow ribbons, known as graphene nanoribbons, it is possible to induce a bandgap. Conventionally the atoms at the edges of nanoribbons are passivated by hydrogen. Recently, single-layer hexagonal boron nitride, which is a wide-bandgap semiconductor, and boron nitride nanoribbons have been studied. Such structures are expected to be produced using a single-layer hexagonal boron nitride as a starting material. The properties of boron nitride nanoribbons are qualitatively different from that of hydrogen-passivated ribbons because of the relatively large ionicities of B and N atoms and the larger energy-gaps of boron nitride. Carbon atoms incorporated in a boron nitride lattice have a stable hexagonal configuration and can form a one-dimensional nanoribbon under suitable chemical potential conditions. It has been shown that armchair nanoribbons embedded in boron nitride sheets are semiconductors. The bandgap opening in these structures is primarily due to the perturbation of the on-site potentials of the edge atoms. The relatively large direct bandgap of graphene nanoribbons embedded in boron nitride renders them as suitable candidates for optoelectronic applications. Furthermore, the energy bandgap of these structures can be tuned with the width of the ribbon, which introduces more flexibility for electronic and optoelectronic applications. We have theoretically studied for the first time the optical properties of armchair graphene nanoribbons embedded in boron nitride.