  The asymptotic structure of outflows from rotating magnetized objects
confined by a uniform external pressure is calculated. The flow is assumed to
be perfect MHD, polytropic, axisymmetric and stationary. The well known
associated first integrals together with the confining external pressure, which
is taken to be independent of the distance to the source, determine the
asymptotic structure. The integrals are provided by solving the flow physics
for the base within the framework of the model developed in Paper I (Lery et
al. 1998), which assumes conical geometry below the fast mode surface, and
ensures the Alfv\'en regularity condition. Far from the source, the outflow
collimate cylindrically. Slow (i.e. with small rotation parameter $\omega$)
rigid rotators give rise to diffuse electric current distribution in the
asymptotic region. They are dominated by gas pressure. Fast rigid rotators have
a core-envelope structure in which a current carrying core is surrounded by an
essentially current free region where the azimuthal magnetic field dominates.
The total asymptotic poloidal current carried away decreases steadily with the
external pressure. A sizeable finite current remains present for fast rotators
even at exceedingly small, but still finite, pressure.
