Solid 4He has been created off the melting curve by growth at a nearly constant mass via the “blocked capillary” technique and growth from the 4He superfluid at constant temperature. The experimental apparatus allows injection of 4He atoms from superfluid directly into the solid. Evidence for the superfluidlike transport of mass through a sample cell filled with hcp solid 4He off the melting curve is found. This mass flux depends on temperature and pressure.