  We use ideal axisymmetric relativistic magnetohydrodynamic simulations to
calculate the spindown of a newly formed millisecond, B ~ 10^{15} G, magnetar
and its interaction with the surrounding stellar envelope during a
core-collapse supernova (SN) explosion. The mass, angular momentum, and
rotational energy lost by the neutron star are determined self-consistently
given the thermal properties of the cooling neutron star's atmosphere and the
wind's interaction with the surrounding star. The magnetar drives a
relativistic magnetized wind into a cavity created by the outgoing SN shock.
For high spindown powers (~ 10^{51}-10^{52} ergs/s), the magnetar wind is
super-fast at almost all latitudes, while for lower spindown powers (~ 10^{50}
erg/s), the wind is sub-fast but still super-Alfvenic. In all cases, the rates
at which the neutron star loses mass, angular momentum, and energy are very
similar to the corresponding free wind values (<~ 30% differences), in spite of
the causal contact between the neutron star and the stellar envelope. In
addition, in all cases that we consider, the magnetar drives a collimated
(~5-10 deg.) relativistic jet out along the rotation axis of the star. Nearly
all of the spindown power of the neutron star escapes via this polar jet,
rather than being transferred to the more spherical SN explosion. The
properties of this relativistic jet and its expected late-time evolution in the
magnetar model are broadly consistent with observations of long duration
gamma-ray bursts (GRBs) and their associated broad-lined Type Ic SN.
