With a half-life of 81.1Myr, 244Pu could be both the heaviest and the shortest-lived nuclide present on Earth as a relic of the last supernova(e) that occurred before the formation of the Solar System. Hoffman [Nature (London)NATUAS0028-083610.1038/234132a0 234, 132 (1971)] reported on the detection of this nuclide (1.0×10-18 g 244Pu/g) in the rare-earth mineral bastnäsite with the use of a mass spectrometer. Up to now these findings were never reassessed. We describe the search for primordial 244Pu in a sample of bastnäsite with the method of accelerator mass spectrometry (AMS). It was performed with a highly sensitive setup, identifying the ions by the determination of their time-of-flight and energy. Using AMS, the stripping to high charge states allows the suppression of any molecular interference. During our measurements we observed no event of 244Pu. Therefore, we can give an upper limit for the abundance of 244Pu in our sample of the mineral bastnäsite of 370 atoms per gram (1.5×10-19 g 244Pu/g). The concentration of 244Pu in our sample of bastnäsite is significantly lower than the previously determined value.