We simulate the evolution of a 10 9 M dark matter halo in a cosmological setting with an adaptive-mesh refinement code as an analogue to local low luminosity dwarf irregular and dwarf spheroidal galaxies. The primary goal of our study is to investigate the roles of reionization and supernova feedback in determining the star formation histories of low mass dwarf galaxies. We include a wide range of physical effects, including metal cooling, molecular hydrogen formation and cooling, photoionization and photodissociation from a metagalactic (but not local) background, a simple prescription for self-shielding, star formation, and a simple model for supernova driven energetic feedback. To better understand the impact of each physical effect, we carry out simulations excluding each major effect in turn. We find that reionization is primarily responsible for expelling most of the gas in our simulations, but that supernova feedback is required to disperse the dense, cold gas in the core of the halo. Moreover, we show that the timing of reionization can produce an order of magnitude difference in the final stellar mass of the system. For our full physics run with reionization at z = 9, we find a stellar mass of about 10 5 M at z = 0, and a mass-to-light ratio within the half-light radius of approximately 130 M /L , consistent with observed low-luminosity dwarfs. However, the resulting median stellar metallicity is 0.06 Z , considerably larger than observed systems. In addition, we find star formation is truncated between redshifts 4 and 7, at odds with the observed late time star formation in isolated dwarf systems but in agreement with Milky Way ultrafaint dwarf spheroidals. We investigate the efficacy of energetic feedback in our simple thermal-energy driven feedback scheme, and suggest that it may still suffer from excessive radiative losses, despite reaching stellar particle masses of about 100 M , and a comoving spatial resolution of 11 pc.