  We present Chandra observations of 17 optically-selected, X-ray weak
narrow-line Seyfert 1 (NLS1) galaxies. These objects were optically identified
by Williams et al. (2002) in the Sloan Digital Sky Survey Early Data Release,
but were not found in the ROSAT All-Sky Survey (RASS) despite having optical
properties similar to RASS-detected NLS1s. All objects in this sample were
detected by Chandra and exhibit a range of 0.5-2 keV photon indices
Gamma=1.1-3.4. One object was not detected in the soft band, but has a best-fit
Gamma=0.25 over the full 0.5-8 keV range. These photon indices extend to values
far below what are normally observed in NLS1s. A composite X-ray spectrum of
the hardest objects in this sample does not show any signs of absorption,
although the data do not prohibit one or two of the objects from being highly
absorbed. We also find a strong correlation between Gamma and L_1keV; this may
be due to differences in L_bol/L_Edd among the NLS1s in this sample. Such
variations are seemingly in conflict with the current paradigm that NLS1s
accrete near the Eddington limit. Most importantly, this sample shows that
strong, ultrasoft X-ray emission is not a universal characteristic of NLS1s; in
fact, a substantial number may exhibit weak and/or low-Gamma X-ray emission.
