  We obtained dual-polarization VLBI observations at six frequencies of the
compact symmetric object J0029+3457 and the CSO candidate J1324+4048. By
comparing the three lower-frequency maps with extrapolations of the high
frequency maps we produced maps of the optical depth as a function of
frequency. The morphology of the optical-depth maps of J1324+4048 is strikingly
smooth, suggestive of a foreground screen of absorbing gas. The spectra at the
intensity peaks fit a simple free-free absorption model, with a reduced chi
square ~ 2, better than a simple synchrotron self-absorption model, in which
the reduced chi square ~ 3.5 - 5.5. We conclude that the case for free-free
absorption in J1324+4048 is strong. The optical-depth maps of J0029+3457
exhibit structure, but the morphology does not correlate with that in the
intensity maps. The fit of the spectra at the peaks to a simple free-free
absorption model yields a reduced chi square ~ 1, but since the turnover is
gradual the fit is relatively insensitive to the input parameters. We find that
free-free absorption by a thin amount of gas in J0029+3457 is likely, but not
definitive. One compact feature in J0029+3457 has an inverted spectrum even at
the highest frequencies. We infer this to be the location of the core and
estimate an upper limit to the magnetic field of order 3 Gauss at a radius of
order 1 pc. In comparison with maps from observations at earlier epochs, no
apparent growth in either J1324+4048 or J0029+3457 is apparent, with upper
limits of 0.03 and 0.02 mas per yr, corresponding to maximum linear separation
speeds of 0.6c and 0.4c.
