s = BoundaryDiscretizeGraphics[Graphics3D[{Ball[]}]];
pos = Flatten[Position[MeshCoordinates[s], {x_, _, _} /; x < 0]];
f = DeleteCases[MeshCells[s, 2][[All, 1]],
x_ /; Intersection[pos, x] != {}];
mesh = MeshRegion[MeshCoordinates[s], Polygon /@ f]