  Abridged: We use three-dimensional SPH simulations to investigate the
collapse of low-mass prestellar cores and the formation and early evolution of
protostellar discs. The initial conditions are slightly supercritical
Bonnor-Ebert spheres in rigid rotation. The core mass and initial radius are
held fixed at M_O=6.1 M_sun and R_O=17,000 AU, and the only parameter that we
vary is the initial angular speed \Omega_O. Protostellar discs forming from
cores with \Omega_O<1.35 10d-13 1/s have radii between 100 and 300 AU and are
quite centrally concentrated; due to heating by gas infall onto the disc and
accretion onto the central object, they are also quite warm, T>100 K, and
therefore stable against gravitational fragmentation. In contrast, more rapidly
rotating cores form discs which are less concentrated and cooler, and have
radii between 400 and 1000 AU; as a consequence they are prone to gravitational
fragmentation and the formation of multiple systems. We derive a criterion that
predicts whether a rigidly rotating core having given M_O, R_O and \Omega_O
will produce a protostellar disc which fragments whilst material is still
infalling from the core envelope. We then apply this criterion to core samples
for which M_O, R_O and \Omega_O have been estimated observationally. We
conclude that the observed cores are stable against fragmentation at this
stage, due to their low angular speeds and the heat delivered at the accretion
shock where the infalling material hits the disc.
