Phytoplankton blooms occur annually at the sea-ice edge throughout the Arctic during the spring melt period. Our study considers how these spring blooms may depend on sea-ice meltwater, focusing on the role of horizontal mixing and advection. We extend the classic Fisher reaction-diffusion equation to consider a time- and space-varying death rate that represents the role of meltwater in the system. Our results indicate that blooms peak at a characteristic distance from the ice edge where (i) meltwater is concentrated enough to stratify the upper ocean such that the phytoplankton are confined near the surface and (ii) phytoplankton have been exposed to sufficient sunlight to allow for optimized growth. The results reproduce key characteristics of a large bloom observed in Fram Strait in May 2019. Our findings support the idea that sea-ice meltwater is of central importance in setting the spatial patterns of Arctic phytoplankton blooms.