The nonlinear saturation of the single‐mode, bump‐on‐tail instability is studied by numerical solutions of the Vlasov equation. Modes close to marginal stability are found to saturate with a field amplitudeE∝&Dgr;2, where the small parameter &Dgr;=2(&ohgr;−&ohgr;’)/&ohgr;’defines the difference between the mode frequency &ohgr; and the frequency &ohgr;’of the marginally stable mode. This scaling agrees with O’Neil’s theory [Phys. Fluids14, 1204 (1971)] of the two‐stream instability, anddoesnotconfirmthe time asymptotic analysis of Simon and Rosenbluth [Phys. Fluids19, 1567 (1976)], which predicts larger amplitudes withE∝&Dgr;1/2.