During eruption, lava domes and flows may become unstable and generate dangerous explosions. Fossil lava-filled eruption conduits and ancient lava flows are often characterized by complex internal variations of gas content. These observations indicate a need for accurate predictions of the distribution of gas content and bubble pressure in an eruption conduit. Bubbly magma behaves as a compressible viscous liquid involving three different pressures: those of the gas and magma phases, and that of the exterior. To solve for these three different pressures, one must account for expansion in all directions and hence for both horizontal and vertical velocity components. We present a new two-dimensional finite element numerical code to solve for the flow of bubbly magma. Even with small dissolved water concentrations, gas overpressures may reach values larger than 1 MPa at a volcanic vent. For constant viscosity the magnitude of gas overpressure does not depend on magma viscosity and increases with the conduit radius and magma chamber pressure. In the conduit and at the vent, there are large horizontal variations of gas pressure and hence of exsolved water content. Such variations depend on decompression rate and are sensitive to the exit boundary conditions for the flow. For zero horizontal shear stress at the vent, relevant to lava flows spreading horizontally at the surface, the largest gas overpressures, and hence the smallest exsolved gas contents, are achieved at the conduit walls. For zero horizontal velocity at the vent, corresponding to a plug-like eruption through a preexisting lava dome or to spine growth, gas overpressures are largest at the center of the vent. The magnitude of gas overpressure is sensitive to changes of magma viscosity induced by degassing and to shallow expansion conditions in conduits with depth-dependent radii. ¿ 2001 American Geophysical Union