A vascular clamp for occluding a blood vessel or duct in a human or animal. The vascular clamp includes a pair of pivoting arms, with a clamping jaw rigidly attached to a distal end of each pivoting arm. A concave substantially semi-cylindrical chamber is formed in each clamping jaw. The clamping jaws are movable between an open position and a closed position, and are aligned so as to form a substantially cylindrical chamber in the closed position. A balloon is mounted in the concave semi-cylindrical chamber of each clamping jaw. Each balloon includes a substantially semi-cylindrical rigid shell conforming to the concave semi-cylindrical chamber and a thin, elastic material pre-filled with a liquid or gaseous fluid at a predetermined pressure. The balloons are configured to completely surround and occlude the blood vessel or duct in the closed position of the vascular clamp. The rigid shell of each balloon is attached to its associated clamping jaw.