A heat dissipation substrate having the maximum value of the coefficient of linear expansion of 10 ppm/K or less in any direction in a plane parallel to the surface within a temperature range from room temperature to 800°C as well as a thermal conductivity of 250 W/m€¢K or higher at 200°C is produced by densifying an alloy composite of CuMo or CuW composed of Cu and coarse powder of Mo or W and subsequently cross-rolling the same alloy composite.