Ion heating by cross-field current-driven ion acoustic turbulence is investigated in a double-plasma device. The spatial heating rate delta U/ delta x of an ion beam is measured as a function of the integrated noise energy density Psi of broadband noise. At the lower values of Psi the observations approximately follow the quasilinear prediction that delta U/ delta x (or delta U/ delta t) varies as Psi tau c where tau c is the autocorrelation time. As Psi approaches its highest observed values, delta U/ delta x reaches saturation. At the transition between the quasilinear and saturation regimes, tau c is of the order of the particle trapping time.