Studies of the distribution of young stars in well-known regions of star formation indicate the existence of a characteristic length scale (~0.04 pc), separating the regime of self-similar clustering from that of binary and multiple systems. The evidence that this length scale is comparable to the size of typical molecular cloud clumps, along with the observed high frequency of companions to pre-main-sequence stars, suggest that stars may ultimately form through fragmentation of collapsing molecular cloud cores. Here we use a new hydrodynamic code to investigate the gravitational collapse and fragmentation of protostellar condensations, starting from moderately centrally condensed (Gaussian), prolate configurations with axial ratios of 2:1 and 4:1 and varying thermal energy (α). All the models start with uniform rotation and ratios of the rotational to the gravitational energy β ~ 0.036. The results indicate that these condensations collapse all the way to form a narrow cylindrical core that subsequently fragments into two or more clumps, even if they are initially close to virial equilibrium (α + β ~ 1/2). The 2:1 clouds formed triple systems for α >~ 0.36 and a binary system for α ~ 0.27, while the 4:1 clouds all formed binary systems independently of α. The mass and separation of the binary fragments increase with increasing the cloud elongation. The widest binaries formed from clouds with α ~ 0.36, and starting from this value, the binary separation decreases with either increasing or decreasing α. In all cases, fragmentation did not result in a net loss of the a/m ratio (specific spin angular momentum per unit mass), as expected from stellar observations. The fragments that formed possess low values of α (~0.06) and are appreciably elongated, and so they could subfragment before becoming true first protostellar cores.