This Letter reports a slow-light fiber Bragg grating (FBG) temperature sensor with a record temperature resolution of ~0.3 m°C √Hz, a drift of only ~1 m°C over the typical duration of a measurement (~30 s), and negligible self-heating. This sensor is particularly useful for applications requiring the detection of very small temperature changes, such as radiation-balanced lasers and the measurement of small absorptive losses using calorimetry. The sensor performance is demonstrated by measuring the heat generated in a pumped Yb-doped fiber. The sensor is also used to measure the slow-light FBG's very weak internal absorption loss (0.02m-1), which is found to be only ~2% of the total loss.