The geothermal data of Lachenbruch et al. <1982> in ice-bearing permafrost at Prudhoe Bay are interpreted in terms of the temperature history on a time scale of 40,000 years or longer. To this end, two theories are developed: a ''forward'' theory to calculate the response of ice-bearing permafrost to a surface temperature disturbance, and an ''inverse'' theory to calculate parameters characterizing the surface temperature history from suitable measurements in this permafrost. The inverse theory shows that two pieces of information characterizing the surface temperature history on a long time scale, and the equilibrium geothermal heat flux, can usually be determined but the results depend upon the quality of the data, the structure of the surface temperature disturbance, and the timing of the measurements; the memory of the permafrost may range from near zero to hundreds of thousands of years. The geothermal data indicate that the Prudhoe Bay ground surface temperature has a 40,000-year average of -10.9¿3.0 ¿C, and a value at glacial maximum of -10.9¿4.6¿C. The limits correspond to the estimated uncertainties in the data, and they depend somewhat on the structure of the surface temperature disturbance over the last 40,000 years or so. The -10.9 ¿C temperature is the same as that which prevailed before the 1.8 K warming of the past century. The limits placed on the variation of the ground surface temperature by this interpretation are not in accord with limits placed by some existing interpretations of other paleoclimatic indicators. It is concluded that a consistent understanding of climate history and geothermal regime in northern Alaska does not yet exist. ¿1991 American Geophysical Union