A bonnet seal assembly for a gate valve includes a generally cylindrical packing nut positioned in the stem bore around the valve stem. The packing nut includes first and second spaced-apart annular sealing surfaces, and the bonnet seal assembly includes a third annular sealing surface connected to or formed integrally with the valve stem and a fourth annular sealing surface formed concentrically in the stem bore. In a first position of the valve stem, the first annular sealing surface sealingly engages the third annular sealing surface to thereby form a first seal between the valve stem and the packing nut, and the second annular sealing surface sealing engages the fourth annular sealing surface to thereby form a second seal between the packing nut and the stem bore. In this manner, the first and second seals together operate to seal the stem bore from fluid pressure in the gate cavity.