As a by-product of high-precision, ultra-deep stellar photometry in the Galactic globular cluster NGC 6397 with the Hubble Space Telescope, we are able to measure a large population of background galaxies whose images are nearly point-like. These provide an extragalactic reference frame of unprecedented accuracy, relative to which we measure the most accurate absolute proper motion ever determined for a globular cluster. We find mu_alpha = 3.56 +/- 0.04 mas/yr and mu_delta = -17.34 +/- 0.04 mas/yr. We note that the formal statistical errors quoted for the proper motion of NGC 6397 do not include possible unavoidable sources of systematic errors, such as cluster rotation. These are very unlikely to exceed a few percent. We use this new proper motion to calculate NGC 6397s UVW space velocity and its orbit around the Milky Way, and find that the cluster has made frequent passages through the Galactic disk.
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