We report the detection of linear polarization from Sgr A* at 750, 850, 1350 and 2000 micron which confirms the contribution of synchrotron radiation. From the lack of polarization at longer wavelengths it appears to arise in the millimetre/sub-millimetre excess. There are large position angle changes between the millimetre and sub-millimetre results and these are discussed in terms of a polarized dust contribution in the sub-millimetre and various synchrotron models. In the model which best explains the data the synchrotron radiation from the excess is self-absorbed in the millimetre region and becomes optically thin in the sub-millimetre. This implies that the excess arises in an extremely compact source 2 Schwarzschild radii.