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We report results from a deep search for redshifted H  i  21 cm absorption in 55 strong Mg  ii λ2796 absorbers (having   W  λ2796  0  &gt; 0.5 Å ) at intermediate redshifts,  0.58 &lt;  z  abs  &lt; 1.70 , with the Green Bank Telescope (GBT) and the Giant Metrewave Radio Telescope (GMRT). Nine detections of H  i  21 cm absorption were obtained, all at  1.17 &lt;  z  abs  &lt; 1.68 , including three systems reported earlier by Gupta et al. (2007). Absorption was not detected at  &gt;3σ  significance in 32 other Mg  ii  absorbers, with 26 of these providing strong upper limits to the H  i  21 cm optical depth,  τ 3σ  &lt; 0.013  per  ∼10 km s −1  . For the latter 26 systems, the spin temperature  T  s  of the absorber must be  &gt;[800 × f ] K  (where  f  is the covering factor), if the H  i  column density is  ≥2 × 10 20  cm −2  , i.e. if the absorber is a damped Lyman α system (DLA). Data on the remaining 13 systems of the sample were affected by radio frequency interference and were hence not useful.  Two of the Mg  ii  absorbers, at   z  abs ∼ 1.4106  towards  2003−025  and at   z  abs ∼ 0.9115  towards 2149+212, are known DLAs. We detect H  i  21 cm absorption towards  2003−025  with the GMRT and estimate the spin temperature of the DLA to be   T  s =[(905 ± 380) × f ] K . Conversely, the GBT observations of 2149+212 resulted in a non‐detection of H  i  21 cm absorption, yielding the 3σ limit   T  s  &gt; [2700 × f ] K .  Excluding ‘associated’ systems (within  3000 km s −1   of the quasar redshift), the detection rate of H  i  21 cm absorption in strong Mg  ii λ2796 absorbers is  per cent, at a 3σ optical depth sensitivity of ∼0.013 per  10 km s −1  . Comparing the detection rates of H  i  21 cm and damped Lyman α absorption in strong Mg  ii  absorber samples yields a detection rate of H  i  21 cm absorption in DLAs of  per cent, consistent with the detection rate in low‐ z  DLAs. Since H  i  21 cm absorption arises in cold neutral gas, this indicates that most gas‐rich galaxies contain significant fractions of cold H  i  by   z ∼ 1 .  Finally, we use the observed detection rate of H  i  21 cm absorption in Mg  ii  absorbers to infer the cosmological mass density of neutral gas in DLAs, assuming that (1) the average H  i  column density in our H  i  21 cm absorber sample is the same as that measured by Rao, Turnshek &amp; Nestor in their DLA sample and (2) the detection rate of H  i  21 cm absorption in the DLAs of our Mg  ii  sample is the same as that in known DLAs at  0.09 &lt;  z  abs  &lt; 3.45 . We obtain  Ω GAS ∼ (0.55 +0.42  −0.22 ) × 10 −3  , at , slightly lower than, but consistent with, the value obtained by Rao, Turnshek &amp; Nestor from their DLA survey at similar redshifts.

A search for H i 21 cm absorption in strong Mg ii absorbers in the redshift desert