We present an analysis of a Chandra ACIS-S observation of the elliptical galaxy NGC 720, to verify the existence of a dark matter halo and to measure its ellipticity. The ACIS-S3 image reveals over 60 point sources distributed throughout the field, most of which were undetected and therefore unaccounted for in previous X-ray studies. For semimajor axes a less than or similar to 150" (18.2 h(70)(-1) kpc), the ellipticity of the di use X-ray emission is consistent with a constant value, epsilon(X) approximate to 0.15, which is systematically less than the values of 0.2-0.3 obtained from previous ROSAT PSPC and HRI observations because of the unresolved point sources contaminating the ROSAT values. The Chandra data confirm the magnitude of the similar to20degrees position angle ( P. A.) twist discovered by ROSAT over this region. However, the twist in the Chandra data is more gradual and occurs at smaller a, also because of the point sources contaminating the ROSAT values. For a greater than or similar to 150" out to a = 185" (22.4 h(70)(-1) kpc), which is near the edge of the S3 CCD, epsilon(X) and P. A. diverge from their values at smaller a. Possible origins of this behavior at the largest a are discussed. Overall, the ellipticities and P. A. twist for a less than or similar to 150 00 can be explained by the triaxial mass model of NGC 720 published by Romanowsky & Kochanek (which could not produce the abrupt P. A. twist in the ROSAT HRI data). Since the optical image displays no substantial isophote twisting, the X-ray P. A. twist requires a massive dark matter halo if the hot gas is in hydrostatic equilibrium. Furthermore, the values of epsilon(X) obtained by Chandra are too large to be explained if the gravitating mass follows the optical light (M proportional to L-*), irrespective of the P. A. twist. The M proportional to L-* hypothesis is inconsistent with the Chandra ellipticities at the 96% confidence level, assuming oblate symmetry, and at the 98% confidence level for prolate symmetry. Thus, both the P. A. twist and the ellipticities of the Chandra image imply the existence of dark matter, independent of the temperature pro le of the gas. This geometric evidence for dark matter cannot be explained by alternative gravity theories, such as the modi cation of Newtonian dynamics (MOND). To constrain the ellipticity of the dark matter halo, we considered both oblate and prolate spheroidal mass models to bracket the full range of ( projected) ellipticities of a triaxial ellipsoid. The dark matter density model, rho proportional to (a(s)(2) + a(2))(-1), provides the best t to the data and gives ellipticities and 1 sigma errors of epsilon = 0.37 +/- 0.03 for oblate and epsilon = 0.36 +/- 0.02 for prolate models. Navarro-Frenk-White (NFW) and Hernquist models give similar ellipticities for the dark matter. These moderate ellipticities for the dark halo are inconsistent with both the nearly spherical halos predicted if the dark matter is self-interacting and the highly flattened halos predicted if the dark matter is cold molecular gas. These ellipticities may also be too large to be explained by warm dark matter, but they are consistent with galaxy-sized halos formed in the currently popular LambdaCDM paradigm.