A cordierite honeycomb structure having a coefficient of thermal conductivity of not less than 0.035 J/cm.multidot..degree.K..multidot.sec, a porosity of not more than 16%, and a coefficient of thermal expansion of not more than 1.0.times.10.sup.-6 /.degree.C. at 40.degree. to 800.degree. C. A process for producing a cordierite honeycomb structures, involves at least the steps of: mixing powders of kaolin, talc, and other starting materials for cordierite together so as to provide a chemical composition of cordierite; and extruding the mixture; and firing the extrudate, wherein the kaolin has an average particle diameter of not more than 5 .mu.m and such a particle size distribution that the proportion of particles with diameters of less than 1 .mu.m is 10% by weight and the talc has an average particle diameter of not more than 10 .mu.m and such a particle size distribution that the proportion of particles with diameters of less than 1 .mu.m is not more than 5%.