  We report results from a deep search for redshifted HI 21cm absorption in 55
strong MgII$\lambda$2796 absorbers (having $W (MgII) > 0.5 \AA$) at
intermediate redshifts, $0.58 < z_{\rm abs} < 1.70$, with the Green Bank
Telescope (GBT) and the Giant Metrewave Radio Telescope (GMRT). Nine detections
of HI 21cm absorption were obtained, all at $1.17 < z_{\rm abs} < 1.68$,
including three systems reported earlier by Gupta et al. (2007). Absorption was
not detected at $> 3\sigma$ significance in 32 other MgII absorbers, with 26 of
these providing strong upper limits to the HI 21cm optical depth,
$\tau_{3\sigma} < 0.013$ per $\sim 10$ km/s. For the latter 26 systems, the
spin temperature $T_s$ of the absorber must be $> [800 \times f]$ K (where $f$
is the covering factor), if the HI column density is $\ge 2 \times 10^{20}$
cm$^{-2}$, i.e. if the absorber is a damped Lyman-$\alpha$ system (DLA). Data
on the remaining 13 systems of the sample were affected by radio frequency
interference and were hence not useful.
  Excluding "associated" systems (within 3000 km/s of the quasar redshift), the
detection rate of HI 21cm absorption in strong MgII absorbers is $x_{\rm
21,MgII} ({\bar z} = 1.1) = 25^{+11}_{-8}$%, at a $3\sigma$ optical depth
sensitivity of $\sim 0.013$ per 10 km/s. Comparing the detection rates of HI
21cm and damped Lyman-$\alpha$ absorption in strong MgII absorber samples
yields a detection rate of HI 21cm absorption in DLAs of $x_{\rm 21,DLA} ({\bar
z} = 1.1) = (73 \pm 27)$%, consistent with the detection rate of HI 21cm
absorption in low-$z$ DLAs. Since HI 21cm absorption arises in cold neutral
gas, this indicates that most gas-rich galaxies contain significant fractions
of cold HI by $z \sim 1$. (abridged)
