Eventually, at great enough distances, almost the entire mass of the galaxy will be enclosed and the rotation curve must become nearly Keplerian and fall off (dark matter, if it is important at all at these scales, would make the curve fall off even faster). Do we have any data that shows the curve becoming nearly Keplerian at great distances? Do we even have an idea of what that distance would be?
Over at astronomy they have a post looking at this from the astronomical perspective. There is no clear boundary and it comes down to arbitrary definitions or decisions.
The rotation curve has been measured out to almost 100kpc. See this article for a nice and current review. From that paper comes this figure:
You can see that we have measurements beyond 100kpc, with the curve clearly decreasing once you go beyond 20 or 30kpc. (Don't think much of the two data points going back up at ~200kpc, that is well within the uncertainty and thus not significant)
For comparison, note that the Andromeda Galaxy is over 700kpc away. Since the two galaxies have similar mass, you could also argue that the Milky Way ends half way between us and there.
Galaxies generally have quite flat rotation curves even at very large distances (much farther away from all the stars forming the galaxy).
At such distances most if the ordinary matter of the galaxy is just a heated gas. The rotation velocity of this gas is measured by observing the Doppler shift of some particular splectral lines.
Astronomers explain these rotation curves by means of a dark matter halo which surrounds the galaxies at large distances.