The viscosity of silicic melts depends strongly on their water content. As bubbles grow in a supersaturated melt, water evaporates from the bubble-melt interface. A diffusive profile develops and leads to steep viscosity gradients across the melt shell. Here we investigate the effects of radial viscosity profiles on the dynamics of bubble growth. We find that the effective melt viscosity resisting gas overpressure in the bubbles is close to the viscosity at the dehydrated rind, and may be higher than that of the surrounding melt by more than an order of magnitude. As a result, bubbles may retain pressures that are higher than ambient pressure for longer times, magma degassing is delayed to shallower depth, and fragmentation of magma due to gas overpressure may occur over a wider range of conditions. Measured water content in eruption products yields information on the average melt viscosity, however additional information about the concentration profiles is needed for estimating the effective viscosity that controlled the evolution of bubble growth in the ejecta.