  Measurements of the intergalactic medium (IGM) temperature provide a
potentially powerful constraint on the reionisation history due to the thermal
imprint left by the photo-ionisation of neutral hydrogen. However, until
recently IGM temperature measurements were limited to redshifts 2 < z < 4.8,
restricting the ability of these data to probe the reionisation history at z >
6. In this work, we use recent measurements of the IGM temperature in the
near-zones of seven quasars at z ~ 5.8 - 6.4, combined with a semi-numerical
model for inhomogeneous reionisation, to establish new constraints on the
redshift at which hydrogen reionisation completed. We calibrate the model to
reproduce observational constraints on the electron scattering optical depth
and the HI photo-ionisation rate, and compute the resulting spatially
inhomogeneous temperature distribution at z ~ 6 for a variety of reionisation
scenarios. Under standard assumptions for the ionising spectra of population-II
sources, the near-zone temperature measurements constrain the redshift by which
hydrogen reionisation was complete to be z > 7.9 (6.5) at 68 (95) per cent
confidence. We conclude that future temperature measurements around other high
redshift quasars will significantly increase the power of this technique,
enabling these results to be tightened and generalised.
