The Muon Accelerator Program (MAP) collaboration is working to develop an ionization cooling channel for muon beams. An ionization cooling channel requires the operation of high-gradient, normal-conducting RF cavities in multi-Tesla solenoidal magnetic fields. However, experiments conducted at Fermilab?s MuCool Test Area (MTA) show that increasing the solenoidal field strength reduces the maximum achievable cavity gradient. This gradient limit is characterized by an RF breakdown process that has caused significant damage to copper cavity interiors. The damage may be caused by field-emitted electrons, focused by the solenoidal magnetic field onto small areas of the inner cavity surface. Local heating may then induce material fatigue and surface damage. Fabricating a cavity with beryllium walls would mitigate this damage due to beryllium?s low density, low thermal expansion, and high electrical and thermal conductivity. We address the design and fabrication of a pillbox RF cavity with beryllium walls, in order to evaluate the performance of high-gradient cavities in strong magnetic fields.