  We report radio continuum observations with the Australia Telescope Compact
Array of two molecular clouds. The impetus for these observations is a search
for synchrotron radiation by cosmic ray secondary electrons/positrons in a
region of enhanced density and possibly high magnetic field. We present
modelling which shows that there should be an appreciable flux of synchrotron
above the more diffuse, galactic synchrotron background.
  The starless core G333.125-0.562 and infrared source IRAS 15596-5301 were
observed at 1384 and 2368 MHz. For G333.125-0.562, we find no significant
levels of radio emission from this source at either frequency, nor any
appreciable polarisation: we place an upper limit on the radio continuum flux
from this source of 0.5 mJy per beam at both 1384 and 2368 MHz. Due to the
higher than expected flux density limits, we also obtained archival ATCA data
at 8640 MHz for this cloud and place an upper limit on the flux density of 50
micro-Jy per beam. Assuming the cosmic ray spectrum is similar to that near the
Sun, and given the cloud's molecular density and mass, we place an upper limit
on the magnetic field of 500 micro-G. IRAS 15596-5301, with an RMS of 50
micro-Jy per beam at 1384 MHz, shows an HII region consistent with optically
thin free-free emission already detected at 4800 MHz. We use the same
prescription as G333 to constrain the magnetic field from this cloud to be less
than 500 micro-G. We find that these values are not inconsistent with the view
that magnetic field values scale with the average density of the molecular
cloud.
