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