Observations of the lengthy tidal streams produced by the destruction of the Sagittarius dwarf spheroidal (Sgr dSph) are capable of providing strong constraints on the shape of the Galactic gravitational potential. However, previous work, based on modeling different stream properties in axisymmetric Galactic models, has yielded conflicting results: while the angular precession of the Sgr leading arm is most consistent with a spherical or slightly oblate halo, the radial velocities of stars in this arm are reproduced only by prolate halo models. We demonstrate that this apparent paradox can be resolved by instead adopting a triaxial potential. Our new Galactic halo model, which simultaneously fits all well-established phase space constraints from the Sgr stream, provides the first conclusive evidence for, and tentative measurement of, triaxiality in an individual dark matter halo. The Milky Way halo within ~60 kpc is best characterized by a minor/major axis ratio of the isovelocity contours c/a 0.67, intermediate/major axis ratio b/a 0.83, and triaxiality parameter T ~ 0.56. In this model, the minor axis of the dark halo is coincident with the Galactic X-axis connecting the Sun and the Galactic center to within ~15°, while the major axis also lies in the Galactic plane, approximately along the Galactic Y-axis.