We construct a hybrid particle/continuum algorithm for linear diffusion in the fluctuating hydrodynamic limit. The particles act as independent random walkers and the fluctuating diffusion equation is solved by a finite difference scheme. At the interface between the particle and continuum computations the coupling is by flux matching, and yields exact mass conservation. This approach is an extension of Adaptive Mesh and Algorithm Refinement [J. Comp. Phys. 154 134 (1999)] to stochastic partial differential equations. We present results from a variety of numerical tests, and in all cases the mean and variance of density are obtained correctly by the stochastic hybrid algorithm. A non-stochastic hybrid (i.e., using only deterministic continuum fluxes) results in the correct mean density, but the variance is diminished except in particle regions away from the interface. Extensions of the approach to other applications are discussed.