We present numerical simulations of axisymmetric, magnetically driven outflows that reproduce the inferred properties of ultrarelativistic gamma-ray burst (GRB) jets. These results extend our previous simulations of outflows accelerated to moderately relativistic speeds, which are applicable to jets of active galactic nuclei. In contrast to several recent investigations, which have employed the magnetodynamics approximation, our numerical scheme solves the full set of equations of special relativistic, ideal magnetohydrodynamics, which enables us to explicitly calculate the jet velocity and magnetic-to-kinetic energy conversion efficiency - key parameters of interest for astrophysical applications. We confirm that the magnetic acceleration scheme remains robust into the ultrarelativistic regime, as previously indicated by semi-analytic self-similar solutions. We find that all current-carrying outflows exhibit self-collimation and consequent acceleration near the rotation axis, but that unconfined outflows lose causal connectivity across the jet and therefore do not collimate or accelerate efficiently in their outer regions. We show that magnetically accelerated jets confined by an external pressure that varies as z-α (0 < α <= 2) assume a paraboloidal shape z ~ ra (where r,z are cylindrical coordinates and a > 1), and we obtain analytic expressions for the one-to-one correspondence between the pressure distribution and the asymptotic jet shape. We demonstrate that the acceleration efficiency of jets with paraboloidal streamlines is >~50 per cent, with the numerical value being higher the lower the initial magnetization. We derive asymptotic analytic expressions for the acceleration of initially cold outflows along paraboloidal streamlines and verify that they provide good descriptions of the simulated flows. Our modelled jets (corresponding to 3/2 < a < 3) attain Lorentz factors Γ >~ 102 on scales ~ 1010-1012cm, consistent with the possibility that long/soft GRB jets are accelerated within envelopes of collapsing massive stars, and Γ >~ 30 on scales ~9 × 108-3 × 1010cm, 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. We also find that Γθv ~ 1 for outflows that undergo an efficient magnetic-to-kinetic energy conversion, where θv is the opening half-angle of the poloidal streamlines. This relation implies that the γ-ray emitting components of GRB outflows accelerated in this way are very narrow, with θv <~ 1° in regions where Γ >~ 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.