In situ permeability and pore pressures were measured 200 m deep beneath the top of the oceanic crust at DSDP site 504B. These measurements have relevance for the transition from convective to conductive heat flow on the south flank of the Costa Rica Rift. Conventional 'slug' and constant rate injection tests were made below a hydraulic packer set at various depths in the hole. The packer was first set in a massive flow until 37 m below the sediment-basement interface. The bulk permeability of the 172.5 m of pillow basalts and basaltic flows below the packer was found to be about 40 millidarcys (4¿10-10 cm2). Measurements over 3- and 15-m intervals at the bottom of the hole in an altered pillow zone indicated a bulk permeability of 2--4 millidarcys. These values are thought to be accurate to ¿30%. Formation pore pressures were found to be approximately 8 bars (~2%) below hydrostatic. Interpretation of the data with respect to simple numerical convection models suggests that the transition from convective to conductive heat flow is controlled by the cessation of convective heat transport through the sedimentary layer rather than the cessation of convection in the sediment. Furthermore, the agreement between observed and modeled underpressure implies that hole 504B penetrated an active ocean crustal convection system. The chick sedimentary layer, layers of basal chert, and massive flow basalts above the layer 2A pillow flows apparently form an impermeable lid, effectively isolating the convection system from the ocean.