  We present a relativistic-MHD numerical study of axisymmetric, magnetically
driven jets with parameters applicable to gamma-ray burst (GRB) flows. We also
present analytic expressions for the asymptotic jet shape and other flow
parameters that agree very well with the numerical results. All
current-carrying outflows exhibit self-collimation and consequent acceleration
near the rotation axis, but unconfined outflows lose causal connectivity across
the jet and therefore do not collimate or accelerate efficiently in their outer
regions. Magnetically accelerated jets confined by an external pressure that
varies with distance with a power-law index < 2 assume a paraboloidal shape and
have an acceleration efficiency > 50%. They attain Lorentz factors > 30 on
scales 10^9-3x10^10 cm, consistent with the possibility that short/hard GRB
jets are accelerated on scales where they can be confined by moderately
relativistic winds from accretion discs, and > 100 on scales 10^10-10^12 cm,
consistent with the possibility that long/soft GRB jets are accelerated within
the envelopes of collapsing massive stars. We also find that the Lorentz factor
of a magnetically accelerated jet is approximately inversely proportional to
the opening half-angle of the poloidal streamlines. This implies that the
gamma-ray emitting components of GRB outflows are very narrow, with a
half-angle < 1 degree in regions where the Lorentz factor exceeds 100, and that
the afterglow light curves of these components would either exhibit a very
early jet break or show no jet break at all.
