The thermal history of cometary nuclei during residence in the Oort cloud is studied with the use of the very low thermal conductivity of amorphous ice recently obtained by Kouchi et al. <1992a>. The heat sources included are (1) radioactive nuclides 40K, 323Th, 235U, and 238U with their chondritic abundances, and (2) latent heat released in transition from amorphous ice to crystalline ice. We model the cometary nucleus as a porous aggregate of grains with each individual grain being composed of a refractory core and an icy mantle. It is assumed that the ice is initially amorphous. The bulk thermal conductivity of a cometary nucleus is assumed to be expressed by the product of the thermal conductivity of individual grains and a reduction factor resulting from the porous structure of the nucleus. Numerical results of the thermal history are presented for various conditions including one case which includes heating by26Al decay. It is shown that the thermal histories are clearly classified into two distinct types depending mainly on the nucleus thermal conductivity &kgr;. (1) Comets with small &kgr; experience a runaway increase in the internal temperature to higher than 120 K during residence in the Oort cloud, in which case most of the ice in the nucleus crystallizes. (2) Comets with a sufficiently large &kgr;, on the other hand, do not exhibit a runaway heating and the temperature is limited to <100 K so that the initial amorphous ice is almost completely preserved. A criterion of nuclear ice crystallization is presented in an analytic expression derived from the analysis of the physical process of the crystallization. A brief discussion is given on the implications of the results for the sources of volatile molecules observed in the coma. ¿ American Geophysical Union 1993