We present a pilot study of the X-ray properties of intermediate-mass (~105-106 M☉) black holes in active galaxies using the Chandra X-ray telescope. Eight of the 10 active galaxies are detected with a significance of at least 3 σ, with X-ray luminosities in the range L0.5-2 keV ≈ 1041-1043 ergs s-1. The optical to X-ray flux ratios are consistent with expectations, given the known correlations between αox and ultraviolet luminosity, while a couple of objects appear to be anomalously X-ray weak. The range of 0.5 to 2 keV photon indices we measure, 1 < Γs < 2.7, is entirely consistent with values found in samples of more luminous sources with more massive black holes. Black hole mass is evidently not a primary driver of soft X-ray spectral index. On the other hand, we do find evidence for a correlation between the X-ray power-law slope and both X-ray luminosity and Eddington ratio, which may suggest that X-ray emission mechanisms weaken at high Eddington ratios. Such a weakening may explain the anomalous X-ray weakness of one of our most optically luminous objects.