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Studies of marine physical, chemical, and microbiological processes benefit from observing in a Lagrangian frame of reference. Some of these processes are related to specific density or temperature ranges. We have developed a method for a Tethys-class long-range autonomous underwater vehicle (LRAUV) (which has a propeller and a buoyancy engine) to track a targeted isothermal layer (within a narrow temperature range) in a stratified water column when operating in buoyancy-controlled drift mode. In this mode, the vehicle shuts off its propeller and autonomously detects the isotherm and stays with it by actively controlling the vehicle's buoyancy. The LRAUV starts on an initial descent to search for the target temperature. Once the temperature falls in the target center bracket, the vehicle records the corresponding depth and adjusts buoyancy to hold that depth. As long as the temperature stays within a tolerance range, the vehicle continues to hold that depth. If the temperature falls out of the tolerance range, the vehicle will increase or decrease buoyancy to reacquire the target temperature and track it. In a June 2015 experiment in Monterey Bay, CA, USA, an LRAUV ran the presented algorithm to successfully track a target isotherm for 13 h. Over the isotherm tracking duration, the LRAUV mostly remained in the 0.5 $^\circ $ C (peak-to-peak) tolerance range as designed, even though the water column's stratification kept changing. This work paves the way to coupling an LRAUV's complimentary modes of flight and drift—searching for an oceanographic feature in flight mode, and then switching to drift mode to track the feature in a Lagrangian frame of reference.