  The theory of the formation of the first stars in the Universe, the so-called
Population III (Pop III), has until now largely neglected the impact of
magnetic fields. Complementing a series of recent studies of the
magneto-hydrodynamic (MHD) aspects of Pop III star formation, we here carry out
a suite of idealized numerical experiments where we ascertain how the
fragmentation properties of primordial protostellar discs are modified if MHD
effects are present. Specifically, starting from cosmological initial
conditions, we focus on the central region in a select minihalo at redshift
$z\sim$ 25, inserting a magnetic field at an intermediate evolutionary stage,
normalized to a fraction of the equipartition value. To explore parameter
space, we consider different field geometries, including uniform, radial,
toroidal, and poloidal field configurations, with the toroidal configuration
being the most realistic. The collapse of the gas is followed for $\sim$8
orders of magnitude in density after the field was inserted, until a maximum of
$10^{15}{\rm \,cm}^{-3}$ is reached. We find that the magnetic field leads to a
delay in the collapse of the gas. Moreover, the toroidal field has the
strongest effect on the collapse as it inhibits the fragmentation of the
emerging disc surrounding the central core and leads to the formation of a more
massive core. The full understanding of the formation of Pop~III stars and
their mass distribution thus needs to take into account the effect of magnetic
fields. We further conclude that ideal MHD is only a first step in this
endeavor, to be followed-up with a comprehensive treatment of dissipative
effects, such as ambipolar diffusion and Ohmic dissipation.
