From Scientific American:
At the low end, some studies find that the galaxy is several hundred billion times as massive as the sun whereas the largest values exceed two trillion solar masses. Astronomers would have an easier task if the galaxy consisted solely of stars. But a huge halo of dark matter engulfs its starry disk and vastly outweighs it. Now remarkable observations of a small galaxy orbiting our own have led to a new number.* * *Combined with the Doppler shift, the proper motion reveals that Leo I orbits the Milky Way at 200 kilometers per second. By comparison, that's nearly as fast as the sun orbits the Milky Way's center, even though the dwarf galaxy is much farther away. Says Boylan-Kolchin, "To sustain a similar velocity at that far a distance requires a lot of extra mass."
How much mass? In a companion study Boylan-Kolchin and his colleagues simulate how giant galaxies such as the Milky Way grow by swallowing lesser galaxies, finding that dwarf galaxies moving as fast as Leo I are almost always bound to the giants, which means Leo I is a true satellite. His team then derives a mass for the Milky Way of 1.6 trillion suns. "That's on the high side," Beers comments. "But it's not outrageous."