It is generally accepted that a large fraction of the earth's volcanism is due to pressure release melting of mantle rock in the ascending limbs of mantle convection cells. A problem that remains largely unsolved is the migration of the melt from the asthenosphere to the earth's surface. We have modeled the early stages of this process by treating the asthenosphere as a porous medium. We assume that mantle rock is moving upward at a constant velocity. The onset of partial melting provides liquid: the liquid creates a liquid-filled porosity, and the interconnected porosity results in permeability. The lighter liquid rises through the solid matrix according to Darcy's law. Some liquid, however, is necessary to provide the necessary permeability. The volume fraction of liquid present is by definition the porosity and is primarily a function of the magma viscosity. Once a critical value is reached, the volume fraction of liquid present does not increase with further partial melting; the liquid produced by melting migrates upward as rapidly as it is produced. At low mantle ascent velocities, less than about 1 mm/yr, the melting takes place over a depth range that lies on the Clapeyron curve. The melt fraction, the magma velocity, and the degree of partial melting within the melting zone are given.