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The temperature profile of a planetary atmosphere is a key diagnostic of radiative and dynamical processes governing the absorption, redistribution, and emission of energy. Observations have revealed dayside stratospheres that either cool [1,2] or warm [3,4] with altitude for a small number of gas giant exoplanets, while others are consistent with constant temperatures [5,6,7,8]. Here we report spectroscopic phase curve measurements for the gas giant WASP-121b,[9] which constrain stratospheric temperatures throughout the diurnal cycle. Variations measured for a water vapor spectral feature reveal a temperature profile that transitions from warming with altitude on the dayside hemisphere to cooling with altitude on the nightside hemisphere. The data are well explained by models assuming chemical equilibrium, with water molecules thermally dissociating at low pressures on the dayside and recombining on the nightside [10,11]. Nightside temperatures are low enough for perovskite (CaTiO3) to condense, which could deplete titanium from the gas phase [12,13] and explain recent non-detections at the day-night terminator [14,15,16,17]. Nightside temperatures are also low enough for refractory species, such as magnesium, iron, and vanadium, to condense. Detections [16,17,18,19] of these metals at the day-night terminator suggest, however, that if they do form nightside clouds, cold trapping is not as effective at removing them from the upper atmosphere. Note: Numbered references have been entered into the "Manuscript Comment" box.