A Multi-gap Resistive Plate Chamber (MRPC) made out of thin float glass is proposed for the outer region of the time of flight (TOF) system for the Compressed Baryonic Matter experiment at FAIR. Usually MRPCs are assembled with ordinary glass plates of 0.5mm or more thickness, but their rate capability is less than the CBM requirement (1.5kHz/cm2). There are two ways to improve the rate capability. The first way is to reduce the bulk resistivity of the glass plates. The second is to reduce the thickness of the glass plates. Tsinghua University has made significant progress in the development of low resistive glass and high rate MRPCs. In this paper we report on three MRPCs produced with float glass plates of 0.7mm, 0.5mm and 0.35mm thickness. Tests with cosmic rays and X-rays were performed at Tsinghua University. The results show that thin float glass MRPCs work well and have the rate capability necessary to meet the demands of the CBM-TOF outer region. Further studies were performed using a continuous 1GeV deuterium beam at the Nuclotron accelerator at the Joint Institute for Nuclear Research (JINR). Time resolution of about 70ps and efficiency higher than 90% were obtained for flux densities up to 3kHz/cm2, exceeding the requirement for the CBM-TOF outer region.