By means of first-principle calculations, we report a stoichiometric crystal structure of BiSb with broken space-inversion symmetry. This structure is insulating in bulk and has nontrivial band topology. We observe a pressure driven Weyl semimetallic electronic phase transition in this BiSb system without a crystal phase change. The obtained Weyl semimetallic phase exists in the 4.0-6.0 GPa pressure range. We find that a total of 12 pairs of Weyl points, 12 monopoles and 12 antimonopoles, exist in the Brillouin zone. Additionally, the spin texture of the bulk BiSb compound appears to be electrically controllable when the interlink between pressure and an electric field is exploited. This produces novel manipulable topological transport properties in this system.