The transient conjugate cooling of a thin horizontal plate enclosed in an adiabatic cavity is studied using asymptotic and numerical techniques. The upper surface of the plate is in contact with a fluid initially at rest. The most important parameter to be obtained is the nondimensional cooling time, which depends on the aspect ratio of the plate E. the nondimensional longitudinal heat conductance of the plate alpha and the Prandtl number of the fluid Pi, for large values of the Rayleigh number of the laminar natural boundary layer flow induced by the density changes. Both the thermally thin (alpha much greater than epsilon(2)) and thick wail regimes (alpha similar to epsilon(2)) are considered in this payer. A minimum nondimensional cooling Lime is achieved for a thin plate (epsilon --> 0) for values of the nondimensional longitudinal thermal conductance of the plate alpha, such as epsilon(2) much less than alpha much less than 1. That is, the process is faster if the longitudinal heat conduction through the wall is negligible.