We report the GMRT detection of HI 21cm absorption from the $z sim 3.39$ damped Lyman-$alpha$ absorber (DLA) towards PKS 0201+113, the highest redshift at which 21cm absorption has been detected in a DLA. The absorption is spread over $sim 115$ km s$^{-1}$ and has two components, at $z = 3.387144 (17)$ and $z = 3.386141 (45)$. The stronger component has a redshift and velocity width in agreement with the tentative detection of Briggs et al. (1997), but a significantly lower optical depth. The core size and DLA covering factor are estimated to be $lesssim 100$ pc and $f sim 0.69$, respectively, from a VLBA 328 MHz image. If one makes the conventional assumption that the HI column densities towards the optical and radio cores are the same, this optical depth corresponds to a spin temperature of $ts sim [(955 pm 160) times (f/0.69)] $ K. However, this assumption may not be correct, given that no metal-line absorption is seen at the redshift of the stronger 21cm component, indicating that this component does not arise along the line of sight to the optical QSO, and that there is structure in the 21cm absorbing gas on scales smaller than the size of the radio core. We model the 21cm absorbing gas as having a two-phase structure with cold dense gas randomly distributed within a diffuse envelope of warm gas. For such a model, our radio data indicate that, even if the optical QSO lies along a line-of-sight with a fortuitously high ($sim 50$%) cold gas fraction, the average cold gas fraction is low, ($lesssim 17%$), when averaged over the the spatial extent of the radio core. Finally, the large mismatch between peak 21cm and optical redshifts and the complexity of both profiles makes it unlikely that the $z sim 3.39$ DLA will be useful in tests of fundamental constant evolution.
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