We present three-dimensional space velocities of stars selected to be consistent with membership in the Virgo stellar substructure. Candidates were selected from SA 103, a single $40’ \times 40’$ field from our proper-motion (PM) survey in Kapteyn’s Selected Areas (SAs), based on the PMs, Sloan Digital Sky Survey (SDSS) photometry, and follow-up spectroscopy of 215 stars. The signature of the Virgo substructure is clear in the SDSS color-magnitude diagram (CMD) centered on SA 103, and 16 stars are identified that have high Galactocentric-frame radial velocities ($V_{\rm GSR} > 50$ km/s) and lie near the CMD locus of Virgo. The implied distance to the Virgo substructure from the candidates is $14 \pm 3$ kpc. We derive mean kinematics from these 16 stars, finding a radial velocity $V_{\rm GSR} = 153 \pm 22$ km/s and proper motions $(\mu_{\alpha {\rm cos} \delta}, \mu_\delta) = (- 5.24, -0.91) \pm (0.43, 0.46)$ mas/yr. From the mean kinematics of these members, we determine that the Virgo progenitor was on an eccentric ($e \sim 0.8$) orbit that recently passed near the Galactic center (pericentric distance $R_{\rm p} \sim 6$ kpc). This destructive orbit is consistent with the idea that the substructure(s) in Virgo originated in the tidal disruption of a Milky Way satellite. N-body simulations suggest that the entire cloud-like Virgo substructure (encompassing the Virgo Overdensity and the Virgo Stellar Stream) is likely the tidal debris remnant from a recently disrupted massive ($\sim 10^9 M_\odot$) dwarf galaxy. The model also suggests that some other known stellar overdensities in the Milky Way halo (e.g., the Pisces Overdensity and debris near NGC 2419 and SEGUE 1) are explained by the disruption of the Virgo progenitor.