We present results from the first attempt to measure strengths of magn
etic fields in dense molecular cloud cores by observations of the Zeem
an effect in the 3 mm lines of CN. The observations were made with the
30 m IRAM radio telescope with a new polarimeter constructed for this
experiment. The two molecular cores which were observed are located a
bout 3' north of Orion-KL (OMC-N4) and about 1' east of IRS 4 in S106
(S106-CN). From CN data for OMC-N4 and S106-CN, respectively, we estim
ate cloud radii of 0.03 and 0.05 pc, masses of 10 and 22 M., and H-2 d
ensities of 1.6 x 10(6) cm(-3). Our CN Zeeman results for the line-of-
sight magnetic fields are B-los(OMC-N4) = +79 +/- 99 mu G and B-los(S1
06-CN) = -57 +/- 199 mu G. Hence, we did not detect magnetic fields. O
ur upper limits for B-los are significantly less than field:strengths
expected for assumption of internal motions at the Alfven speed, of a
critical mass to magnetic flux ratio, or of virial equilibrium, but th
e very small sample of clouds and the uncertainties in the \B\(Alfven)
, \B\(critical), and \B\(virial) estimates preclude definite conclusio
ns.