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 (0.02 m-1), which is found to be only ∼2% of the total loss.