  Long duration Gamma-Ray Bursts (GRBs) originate from the core collapse of
massive stars, but the identity of the central engine remains elusive. Previous
work has shown that rapidly spinning, strongly magnetized proto-neutron stars
(`millisecond proto-magnetars') produce outflows with energies, timescales, and
magnetizations sigma_0 (maximum Lorentz factor) that are consistent with those
required to produce long GRBs. Here we extend this work in order to construct a
self-consistent model that directly connects the properties of the central
engine to the observed prompt emission. Just after the launch of the supernova
shock, a wind heated by neutrinos is driven from the proto-magnetar. The
outflow is collimated into a bipolar jet by its interaction with the star. As
the magnetar cools, the wind becomes ultra-relativistic and Poynting-flux
dominated (sigma_0 >> 1) on a timescale comparable to that required for the jet
to clear a cavity through the star. Although the site and mechanism of the
prompt emission are debated, we calculate the emission predicted by two models:
magnetic dissipation and internal shocks. Our results favor the magnetic
dissipation model in part because it predicts a relatively constant `Band'
spectral peak energy E_peak with time during the GRB. The jet baryon loading
decreases abruptly when the neutron star becomes transparent to neutrinos at t
~ 10-100 seconds. Jets with ultra-high magnetization cannot effectively
accelerate and dissipate their energy, suggesting this transition ends the
prompt emission and may explain the steep decay phase that follows. We assess
several phenomena potentially related to magnetar birth, including low
luminosity GRBs, thermal-rich GRBs/X-ray Flashes, very luminous supernovae, and
short duration GRBs with extended emission.
