  In this paper, we consider the cubic nonlinear Schr\"odinger equation with
third order dispersion on the circle. In the non-resonant case, we prove that
the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac
34$, are quasi-invariant under the flow. In establishing the result, we apply
gauge transformations to remove the resonant part of the dynamics and use
invariance of the Gaussian measures under these gauge transformations.
