We study 
preconditioning for the saddle-point systems that arise
in a stochastic Galerkin mixed formulation of the steady-state diffusion
problem with random data. The key ingredient is a multigrid V-cycle for a
weighted, stochastic
operator, acting on a certain tensor product space of random
fields with finite variance. The traditional deterministic
Arnold-Falk-Winther multigrid algorithm is exploited by varying the
spatial discretization from grid to grid whilst keeping the stochastic
discretization fixed. We extend the deterministic analysis to accommodate
the modified operator and establish spectral equivalence bounds
with a new multigrid V-cycle operator that are independent of the
discretization parameters. We then implement multigrid within a
block-diagonal preconditioner for the full stochastic saddle-point
problem, summarize eigenvalue bounds for the preconditioned system
matrices and investigate the impact of all the discretization parameters
on the convergence rate of preconditioned MINRES.