XENON10 is an experiment to directly detect weakly interacting massive particle (WIMPs), which may comprise the bulk of the non-baryonic dark matter in our Universe. We report new results for spin-dependent WIMP-nucleon interactions with 129 Xe and 131 Xe from 58.6 live-days of operation at the Laboratori Nazionali del Gran Sasso (LNGS). Based on the non-observation of a WIMP signal in 5.4 kg of fiducial liquid xenon mass, we exclude previously unexplored regions in the theoretically allowed parameter space for neutralinos. We also exclude a heavy Majorana neutrino with a mass in the range of ∼10 GeV/c 2 -2 TeV/c 2 as a dark matter candidate under standard assumptions for its density and distribution in the galactic halo.PACS numbers: 95.35.+d, 29.40.Mc, 95.55.Vj Evidence for a significant cold dark matter component in our universe is stronger than ever [1, 2, 3], a wellmotivated particle candidate being the lightest neutralino from super-symmetric extensions to the Standard Model [4]. Such a particle is neutral, non-relativistic, stable, and more generally classified as a Weakly Interacting Massive Particle (WIMP). The open question of the nature of WIMPs is being addressed by numerous direct and indirect detection experiments [5,6,7].Among these, the XENON10 experiment aims to directly detect galactic WIMPs scattering elastically from Xe atoms. Moving with velocities around 10 −3 c, WIMPs can couple to nucleons via both spin-independent and spin-dependent (axial vector) interactions. Spinindependent WIMP-nucleon couplings are in general smaller than axial vector couplings [4]. However, for low momentum transfer, they benefit from coherence across the nucleus, and therefore the overall event rate for WIMP interactions is expected to be dominated by the spin-independent coupling for target nuclei with A≥30. The sensitivity of XENON10 to spin-independent interactions is published in [8].We report here on a spin-dependent analysis of 58.6 live-days of WIMP-search data, taken in low-background conditions at LNGS, which provides ∼3100 meters water equivalent rock overburden. XENON10 is a dual phase (liquid and gas) xenon time projection chamber, discriminating between the predominantly electron-recoil background and the expected nuclear-recoil WIMP signal via the distinct ratio of ionization to scintillation for each type of interaction [8]. A nuclear recoil energy threshold of 4.5 keV was achieved, and 10 candidate events were recorded for an exposure of about 136 kg days after analysis cuts (the fiducial mass was 5.4 kg). Although all observed events are consistent with expected background from electron recoils (see [8] for details on the analysis and the candidate events), no background subtraction is employed for calculating the WIMP upper limits. In the following analysis, we use identical data quality, fiducial volume, and physics cuts as reported in [8].For axial WIMP-nuclei interactions, the WIMPs couple to the spins of the nucleons. Although the interaction with the nucleus is coherent (as it is in the s...