  We use the technique of chopped nonlinear magneto-optic rotation (NMOR) in a
room temperature $^{133}$Cs vapor cell to measure the permanent electric dipole
moment (EDM) in the atom. The cell has paraffin coating on the walls to
increase the relaxation time. The signature of the EDM is a shift in the Larmor
precession frequency which is correlated with the application of an E field. We
analyze errors in the technique, and show that the main source of systematic
error is the appearance of a longitudinal B field when the E field is applied.
This error can be eliminated by doing measurements on the two ground hyperfine
levels. Using an E field of 2.6 kV/cm, we place an upper limit on the electron
EDM of $ 2.9 \times 10^{-22} $ e-cm ($95 \%$ confidence). This limit can be
increased by 7 orders-of-magnitude---and brought below the current best
experimental value---with easily implementable improvements to the technique.
