  We present the first analytic study of finite-size effects on critical
diffusion above and below T_c of three-dimensional Ising-like systems whose
order parameter is coupled to a conserved density. We also calculate the
finite-size relaxation time that governs the critical order-parameter
relaxation towards a metastable equilibrium state below T_c. Two new universal
dynamic amplitude ratios at T_c are predicted and quantitative predictions of
dynamic finite-size scaling functions are given that can be tested by
Monte-Carlo simulations.
