The energy-sensitive detection of heavy ions with calorimetric low temperature detectors was investigated in the energy range of E=0.1-1 MeV/amu, commonly used for accelerator mass spectrometry (AMS). The detectors used consist of sapphire absorbers and superconducting aluminum transition edge thermometers operated at T approximately 1.5 K. They were irradiated with various ion beams (13C, 197Au, 238U) provided by the VERA tandem accelerator in Vienna, Austria. The relative energy resolution obtained was DeltaE/E=(5-9) x 10(-3), even for the heaviest ions such as 238U. In addition, no evidence for a pulse height defect was observed. This performance allowed for the first time to apply a calorimetric low temperature detector in an AMS experiment. The aim was to precisely determine the isotope ratio of 236U/238U for several samples of natural uranium, 236U being known as a sensitive monitor for neutron fluxes. Replacing a conventionally used detection system at VERA by the calorimetric detector enabled to substantially reduce background from neighboring isotopes and to increase the detection efficiency. Due to the high sensitivity achieved, a value of 236U/238U=6.1 x 10(-12) could be obtained, representing the smallest 236U/238U ratio measured at the time. In addition, we contributed to establishing an improved material standard of 236U/238U, which can be used as a reference for future AMS measurements.