ABSTRACT We report a deep Giant Metrewave Radio Telescope search for Galactic H i 21-cm absorption towards the quasar B0438−436, yielding the detection of wide, weak H i 21-cm absorption, with a velocity-integrated H i 21-cm optical depth of 0.0188 ± 0.0036 km s−1. Comparing this with the H i column density measured in the Parkes Galactic All-Sky Survey gives a column density-weighted harmonic mean spin temperature of 3760 ± 365 K, one of the highest measured in the Galaxy. This is consistent with most of the H i along the sightline arising in the stable warm neutral medium. The low-peak H i 21-cm optical depth towards B0438−436 implies negligible self-absorption, allowing a multi-Gaussian joint decomposition of the H i 21-cm absorption and emission spectra. This yields a gas kinetic temperature of $T_{\rm k} \le (4910 \pm \lt t\gt 1900)$ K, and a spin temperature of $T_{\rm s} = (1000 \pm 345)$ K for the gas that gives rise to the H i 21-cm absorption. Our data are consistent with the H i 21-cm absorption arising from either the stable WNM, with $T_{\rm s} \ll T_{\rm k}$, $\rm T_k \approx 5000$ K, and little penetration of the background Lyman-α radiation field into the neutral hydrogen, or the unstable neutral medium, with $T_{\rm s} \approx T_{\rm k} \approx 1000\,{\rm K}$.