Using an apparatus that allows superfluid liquid 4_He to be in contact with hcp solid 4_He at pressures greater than the bulk melting pressure of the solid, we have performed experiments that show evidence for 4_He mass flux through the solid and the likely presence of superfluid inside the solid. We present results that show that a thermomechanical equilibrium in quantitative agreement with the fountain effect exists between two liquid reservoirs connected to each other through two superfluid-filled Vycor rods in series with a chamber filled with solid〖 4〗_He. We use the thermomechanical effect to induce flow through the solid and measure the flow rate. On cooling, mass flux appears near T = 600 mK and rises smoothly as the temperature is lowered. Near T = 75 mK a sharp drop in the flux is present. The flux increases as the temperature is reduced below mK. We comment on possible causes of this flux minimum.