We report exciton spin memory in a single InAs(0.25)P(0.75) quantum dot embedded in an InP nanowire. By synthesizing clean quantum dots with linewidths as narrow as about 30 microeV, we are able to resolve individual spin states at magnetic fields on the order of 1 T. We can prepare a given spin state by tuning excitation polarization or excitation energy. These experiments demonstrate the potential of this system to form a quantum interface between photons and electrons.