The electronic structure and structural stability of LaBiPt are studied by using relativistic local-density energy band calculations. Total-energy calculations for three different atomic configurations within the MgAgAs-type crystal structure show that the Pt atom is definitely on a unique site. For the stable configuration, LaBiPt is semimetallic with hole and electron carrier concentrations of 10-4e. Calculated hole Fermi surfaces have nearly a cubic shape, whose cross sections and their field-angle dependence are in good agreement with recently observed Shubnikov-de Haas result. LaBiPt becomes a zero-gap semiconductor without the spin-orbit coupling, indicating that LaBiPt is a spin-orbit induced semimetal. The semimetallic nature originates mostly in BiPt and may be inherent in wider material systems like RBiPt with rare-earth elements R.