  We show, by means of a perturbative weakly nonlinear analysis, that the
axisymmetric magneto-rotational instability (MRI) in a magnetic Taylor-Couette
(mTC) flow in a thin-gap gives rise, for very small magnetic Prandtl numbers
(P_m), to a real Ginzburg-Landau equation for the disturbance amplitude. The
saturation amplitude A_s is found to scale in this regime as P^\delta, with
1/2<\delta<2/3 (depending on the boundary conditions adopted). The asymptotic
results are shown to comply with numerical calculations performed by using a
spectral code. They suggest that the transport due to the nonlinearly developed
MRI may be vanishingly small for P_m << 1.
