  We aim to measure the proper motion along the Sagittarius stream that is the
missing piece to determine its full 6D phase space coordinates. We conduct a
blind search of over-densities in proper motion from Gaia DR2 in a broad region
around the Sagittarius stream by applying wavelet transform techniques. We find
that for most of the sky patches, the highest intensity peaks delineate the
path of the Sagittarius stream. The 1500 peaks identified depict a continuous
sequence spanning almost $2\pi$ in the sky, only obscured when the stream
crosses the Galactic disk. Altogether, around $100\,000$ stars potentially
belong to the stream as indicated by a coarse inspection of the
colour-magnitude diagrams. From these stars, we determine the proper motion
along the Sagittarius stream, making it the proper motion sequence with the
largest span and continuity ever measured for a stream. A first comparison with
existing N-body models of the stream reveals some discrepancies, especially
near the pericentre of the trailing arm and an overestimation of the total
proper motion for the leading arm. Our study can be the starting point for
determining the variation of the population of stars along the stream, the
distance to the stream with red clump stars, and the solar motion. It will also
allow a much better measurement of the Milky Way potential.
