The dissipative infall of gas during the formation of a galaxy modifies the density profile and shape of the dark halo. Gas dissipates energy radiatively and sinks to the center of the dark halo forming the luminous part of a galaxy. The resulting central density enhancement can alter the halo's orbital distribution. We simulate dissipative infall inside of an initially triaxial N-body dark halo by slowly growing a potential in the center of the particle distribution. The dark halo transforms from a prolate-triaxial halo ($T \sim 0.8$) to an oblate-triaxial halo ($T \sim 0.5$) while approximately preserving the flattening ($c/a \sim 0.5$). The main implication is that dark halos are rounder and more oblate than previous predictions of purely collisionless simulations with the new constraint that $b/a \gapp 0.7$. At the same time, the distribution of intrinsic flattenings of dark halos ($\langle c/a \rangle = 0.5$, $\langle (c/a)^2 \rangle ^{1/2} = 0.15$) is preserved during the period of baryonic infall. The oval distortions of disk galaxies should therefore be slightly less than original predictions from collisionless dark halos. The predicted distribution of shapes for dark halos from cosmological N-body simulations is in better agreement with distribution of ellipticities of elliptical galaxies if we assume all halos and galaxies become oblate-triaxial in response to baryonic dissipation. The observed distribution of kinematic misalignment angles is also consistent with this shape distribution if

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