(Abridged) We present an analysis of a Chandra ACIS-S observation of the elliptical galaxy NGC 720 to verify the existence of a dark matter (DM) halo and to measure its ellipticity. The ACIS-S3 image reveals over 60 point sources. For semi-major axes a<~150 (18.2h_{70}^{-1} kpc) the ellipticity of the diffuse emission is ex ~0.15, which is less than the values 0.2-0.3 obtained from ROSAT because the point sources contaminated the ROSAT values. The Chandra data confirm the ~20 deg position angle (PA) twist discovered by ROSAT, but the Chandra twist is more gradual also because of the point sources contaminating the ROSAT values. Overall the ex and PA values for a<~150 can be explained by the triaxial model of NGC 720 published by Romanowsky & Kochanek. Since the optical image displays no substantial isophote twisting, the X-ray PA twist requires a massive DM halo if the hot gas is in hydrostatic equilibrium. The mass-follows-light hypothesis is also inconsistent with the Chandra ellipticities at the 96% (98%) level for oblate (prolate) symmetry. Thus, both the PA twist and the ellipticities of the Chandra image imply a DM halo independent of the gas T profile -- evidence that cannot be obviated by alternative gravity theories such as MOND. The DM density model, rho ~a^{-2}, provides the best fit and gives ellipticities of 0.37 +/- 0.03 (0.36 +/- 0.02) for oblate (prolate) models. These moderate ellipticities for the DM halo are inconsistent with both the nearly spherical halos predicted if the DM is self-interacting and with the highly flattened halos predicted if the DM is cold molecular gas. These ellipticities may also be too large to be explained by warm DM, but are consistent with galaxy-sized halos formed in the Lambda-CDM paradigm.
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