The transverse stability of a relativistic electron beam focused by an ion channel in the presence of a diffuse background plasma is investigated. The linear behavior of the ion‐hose and electron two‐stream instabilities is treated analytically using a spread‐mass model for the beam and ion channel and a cold‐fluid model for the plasma. The electron two‐stream instability is found to be quite weak. As the plasma neutralization radius approaches the beam radius, the ion‐hose growth rate is reduced up to 50% before the model’s assumptions break down. Particle‐in‐cell simulations confirm the linear analytic theory and show that the electron two‐stream instability can saturate nonlinearly with little beam emittance growth.