  Numerical hydrodynamics calculations are performed to determine conditions
under which giant planet eccentricities can be excited by parent gas disks.
Unlike in other studies, Jupiter-mass planets are found to have their
eccentricities amplified --- provided their orbits start eccentric. We
disentangle the web of co-rotation, co-orbital, and external resonances to show
that this finite-amplitude instability is consistent with that predicted
analytically. Ellipticities can grow until they reach of order the disk's
aspect ratio, beyond which the external Lindblad resonances that excite
eccentricity are weakened by the planet's increasingly supersonic epicyclic
motion. Forcing the planet to still larger eccentricities causes catastrophic
eccentricity damping as the planet collides into gap walls. For standard
parameters, the range of eccentricities for instability is modest; the
threshold eccentricity for growth ($\sim$$0.04$) is not much smaller than the
final eccentricity to which orbits grow ($\sim$$0.07$). If this threshold
eccentricity can be lowered (perhaps by non-barotropic effects), and if the
eccentricity driving documented here survives in 3D, it may robustly explain
the low-to-moderate eccentricities $\lesssim 0.1$ exhibited by many giant
planets (including Jupiter and Saturn), especially those without planetary or
stellar companions.
