A cryosurgical instrument including: a shaft having a closed distal end defining an expansion chamber and an open proximal end that receives an inflow of cryogen and to exhaust a flow of expanded cryogen; and a heat exchanger with a cryogen delivery tube that spirals around a core disposed along longitudinal axis thereof. The cryogen delivery tube, at least where it spirals, is in fluid tight contact with the inner surface of the shaft so as to form spiraling cryogen exhaust pathways that extend along a portion of a length of the cryosurgical instrument from the distal end of the shaft. The cryogen delivery tube forces a two-phase flow of cryogen to undergo phase separation by centrifugal forces and urges the liquid phase against the external walls of the spiral coil and the outer surface of the spiral gap. Heat exchanging, through boiling, occurs close to the external wall of the instrument.