I report the detection of H i 21cm absorption in two high column density damped Lyman- \alpha absorbers ( DLAs ) at z \approx 2 using new wide-band 250 - 500 MHz receivers onboard the Giant Metrewave Radio Telescope . The integrated H i 21cm optical depths are 0.85 \pm 0.16 km s ^ { -1 } ( TXS1755+578 ) and 2.95 \pm 0.15 km s ^ { -1 } ( TXS1850+402 ) . For the z = 1.9698 DLA towards TXS1755+578 , the difference in H i 21cm and C i profiles and the weakness of the radio core suggest that the H i 21cm absorption arises towards radio components in the jet , and that the optical and radio sightlines are not the same . This precludes an estimate of the DLA spin temperature . For the z = 1.9888 DLA towards TXS1850+402 , the absorber covering factor is likely to be close to unity , as the background source is extremely compact , with all the 5 GHz emission arising from a region of size \leq 1.4 mas . This yields a DLA spin temperature of { T _ { s } } = ( 372 \pm 18 ) \times ( f / 1.0 ) K , lower than typical { T _ { s } } values in high- z DLAs . This low spin temperature and the relatively high metallicity of the z = 1.9888 DLA ( [ Zn/H ] = ( -0.68 \pm 0.04 ) ) are consistent with the anti-correlation between metallicity and spin temperature that has been earlier found in damped Lyman- \alpha systems .