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The neutralino, the lightest stable supersymmetric particle, is a strongtheoretical candidate for the missing astronomical ``dark matter''. A profusionof such neutralinos can accumulate near the Sun when they lose energy uponscattering and are gravitationally captured. Pair-annihilations of thoseneutralinos may produce very high energy (VHE, above $100 GeV$) gamma-rays.  Milagro is an air shower array which uses the water Cherenkov technique todetect extensive air showers and is capable of observing VHE gamma-rays fromthe direction of the Sun with an angular resolution of $0.75^{\circ}$. Analysisof Milagro data with an exposure to the Sun of 1165 hours presents the firstattempt to detect TeV gamma-rays produced by annihilating neutralinos capturedby the Solar system and shows no statistically significant signal. Resultinglimits that can be set on gamma-ray flux due to near-Solar neutralinoannihilations and on neutralino cross-section are presented

Search for very high energy gamma rays from WIMP annihilations near the Sun with the Milagro detector