We present results from Chandra observations of 14 ultraluminous infrared galaxies [ULIRGs; log ≥ 12] with redshifts between 0.04 and 0.16. The goals of the observations were to investigate any correlation between infrared color or luminosity and the properties of the X-ray emission and to attempt to determine whether these objects are powered by starbursts or active galactic nuclei (AGNs). The sample contains approximately the same number of high- and low-luminosity objects and "warm" and "cool" ULIRGs. All 14 galaxies were detected by Chandra. Our analysis shows that the X-ray emission of the two Seyfert 1 galaxies in our sample is dominated by an AGN. The remaining 12 sources are too faint for conventional spectral fitting to be applicable. Hardness ratios were used to estimate the spectral properties of these faint sources. The photon indices, Γ-values, for our sample plus the Chandra-observed 2003 sample from Ptak et al. peak in the range 1.0-1.5, consistent with expectations for X-ray binaries in a starburst, an absorbed AGN, or hot bremsstrahlung from a starburst or AGN. The values of Γ for the objects in our sample classified as Seyferts (type 1 or 2) are larger than 2, while those classified as H II regions or LINERs tend to be less than 2. The hard X-ray to far-infrared ratios for the 12 weak sources are similar to those of starbursts, but we cannot rule out the possibility of absorbed, possibly Compton-thick, AGNs in some of these objects. Two of these faint sources were found to have X-ray counterparts to their double optical and infrared nuclei.