  We present new measurements of the thermal state of the intergalactic medium
(IGM) at $z\sim2.4$ derived from absorption line profiles in the Ly$\alpha$
forest. We use a large set of high-resolution hydrodynamical simulations to
calibrate the relationship between the temperature-density ($T$--$\Delta$)
relation in the IGM and the distribution of HI column densities, $N_{\rm HI}$,
and velocity widths, $b_{\rm HI}$, of discrete Ly$\alpha$ forest absorbers.
This calibration is then applied to the measurement of the lower cut-off of the
$b_{\rm HI}$--$N_{\rm HI}$ distribution recently presented by Rudie et al.
(2012). We infer a power-law $T$--$\Delta$ relation,
$T=T_{0}\Delta^{\gamma-1}$, with a temperature at mean density,
$T_{0}=[1.00^{+0.32}_{-0.21}]\times10^{4}\rm\,K$ and slope
$(\gamma-1)=0.54\pm0.11$. The slope is fully consistent with that advocated by
the analysis of Rudie et al (2012); however, the temperature at mean density is
lower by almost a factor of two, primarily due to an adjustment in the
relationship between column density and physical density assumed by these
authors. These new results are in excellent agreement with the recent
temperature measurements of Becker et al. (2011), based on the curvature of the
transmitted flux in the Ly$\alpha$ forest. This suggests that the thermal state
of the IGM at this redshift is reasonably well characterised over the range of
densities probed by these methods.
