We report on an experimental study of the mixing of two miscible fluids in a porous medium driven by either a gravitationally stable or unstable linear displacement flow. A solution of a given NaCl concentration was injected at a constant rate into a homogeneous bead pack saturated with an aqueous solution of different NaCl concentration. The position and width of the fluid interface were recorded using digital image analysis. Our results highlight the strong interplay between gravity and mechanical dispersion. For stable flows, at low P¿clet number Pe, the dispersion coefficient increases with Pe, whereas for Pe >500, it becomes constant. Although similar regimes are observed with pure mixing, the stabilizing effect of gravity reduces significantly the dispersion coefficient by a factor of 2--4, even for density contrasts as low as 0.07%. For unstable flows, gravitational instability is delayed by mechanical dispersion. The ratio of the gravitational to background fluid velocity, G, controls the initial diffusive growth of the mixing zone and the time at which the instability subsequently arises. The wavelength of the instability scales with the width of the diffusive mixing zone at onset of instability and its subsequent linear convective growth rate is one fourth of the unstable gravity speed. For natural porous flows, mechanical dispersion can suppress the onset of gravitational instability until the mixing zone is 1--2 m thick. The width of this mixed zone then controls the length scale of the unstable fingers.