We present an analysis of the Swift Burst Alert Telescope (BAT) and X-ray telescope (XRT) data of GRB060602B, which is most likely an accreting neutron star in a binary system and not a gamma-ray burst. Our analysis shows that the BAT burst spectrum is consistent with a thermonuclear flash (type I X-ray burst) from the surface of an accreting neutron star in a binary system. The X-ray binary nature is further confirmed by the report of a detection of a faint point source at the position of the XRT counterpart of the burst in archival XMM-Newton data approximately six year before the burst and in more recent XMM-Newton data obtained at the end of 2006 September (nearly four months after the burst). Since the source is very likely not a gamma-ray burst, we rename the source Swift J1749.4-2807, based on the Swift/BAT discovery coordinates. Using the BAT data of the type I X-ray burst, we determined that the source is at most at a distance of 6.7 +/- 1.3 kpc. For a transiently accreting X-ray binary, its soft X-ray behaviour is atypical: its 2-10 keV X-ray luminosity (as measured using the Swift/XRT data) decreased by nearly three orders of magnitude in about 1 day, much faster than what is usually seen for X-ray transients. If the earlier phases of the outburst also evolved this rapidly, then many similar systems might remain undiscovered because the X-rays are difficult to detect and the type I X-ray bursts might be missed by all the sky surveying instruments. This source might be part of a class of very fast transient low-mass X-ray binary systems of which there may be a significant population in our Galaxy.