We present new measurements of the temperature of the intergalactic medium (IGM) derived from the Lyα forest over 2.0 ≤z≤ 4.8. The small-scale structure in the forest of 61 high-resolution quasi-stellar object spectra is quantified using a new statistic, the curvature, and the conversion to temperature calibrated using a suite of hydrodynamic simulations. At each redshift, we focus on obtaining the temperature at an optimal overdensity probed by the Lyα forest, , where the temperature is nearly a one-to-one function of the curvature regardless of the slope of the temperature–density relation. The median 2σ statistical uncertainty in these measurements is 8 per cent, though there may be comparable systematic errors due to the unknown amount of Jeans smoothing in the IGM. We use our results to infer the temperature at the mean density, T0. Even for a maximally steep temperature–density relation, T0 must increase from ∼8000 K at z≃ 4.4 to ≳12 000 K at z≃ 2.8. This increase is not consistent with the monotonic decline in T0 expected in the absence of He ii reionization. We therefore interpret the observed rise in temperature as evidence of He ii reionization beginning at z≳ 4.4. The evolution of T0 is consistent with an end to He ii reionization at z∼ 3, as suggested by opacity measurements of the He ii Lyα forest, although the redshift at which T0 peaks will depend somewhat on the evolution of the temperature–density relation. These new temperature measurements suggest that the heat input due to the reionization of He ii dominates the thermal balance of the IGM over an extended period with Δz≳ 1.