A gas-mixture-ignition-time estimation apparatus for an internal combustion engine which comprises premixed-gas-mixture-autoignition-start-time estimation means for estimating a time at which a premixed gas mixture previously formed through mixing of a cylinder interior gas and fuel starts autoignition upon compression by generation of a hot flame, the cylinder interior gas being a gas within a combustion chamber, and the fuel being fuel for premixed-charge compression ignition combustion injected into the combustion chamber earlier than a near compression top dead center and being different from fuel for diffusion combustion injected into the combustion chamber at the near compression top dead center, compression-start-time cylinder-interior-gas-state-quantity acquisition means for acquiring a state quantity of the cylinder interior gas at the time of start of compression of the cylinder interior gas, the state quantity including temperature, pressure, and internal energy of the cylinder interior gas; compression-attributable-cylinder-interior-gas-state-quantity-change-amount estimation means for estimating the amount of a change in the state quantity of the cylinder interior gas attributable to compression of the cylinder interior gas; the gas-mixture-ignition-time estimation apparatus being characterized by comprising cool-flame-heat-generation-quantity estimation means for estimating a cool-flame-heat-generation-quantity-corresponding value, which is a value corresponding to a heat generation quantity of a cool flame generated prior to autoignition of the premixed gas mixture, wherein the premixed-gas-mixture-autoignition-start-time estimation means estimates the time at which the premixed gas mixture starts autoignition, on the basis of at least the acquired cylinder interior gas state quantity at the time of start of compression, the estimated compression-attributable interior gas state quantity change amount, and the estimated cool-flame-heat-generation-quantity-corresponding value. premixed-gas-mixture-autoignition-start-time estimation means for estimating a time at which a premixed gas mixture previously formed through mixing of a cylinder interior gas and fuel starts autoignition upon compression by generation of a hot flame, the cylinder interior gas being a gas within a combustion chamber, and the fuel being fuel for premixed-charge compression ignition combustion injected into the combustion chamber earlier than a near compression top dead center and being different from fuel for diffusion combustion injected into the combustion chamber at the near compression top dead center, compression-start-time cylinder-interior-gas-state-quantity acquisition means for acquiring a state quantity of the cylinder interior gas at the time of start of compression of the cylinder interior gas, the state quantity including temperature, pressure, and internal energy of the cylinder interior gas; compression-attributable-cylinder-interior-gas-state-quantity-change-amount estimation means for estimating the amount of a change in the state quantity of the cylinder interior gas attributable to compression of the cylinder interior gas; the gas-mixture-ignition-time estimation apparatus being characterized by comprising cool-flame-heat-generation-quantity estimation means for estimating a cool-flame-heat-generation-quantity-corresponding value, which is a value corresponding to a heat generation quantity of a cool flame generated prior to autoignition of the premixed gas mixture, wherein the premixed-gas-mixture-autoignition-start-time estimation means estimates the time at which the premixed gas mixture starts autoignition, on the basis of at least the acquired cylinder interior gas state quantity at the time of start of compression, the estimated compression-attributable interior gas state quantity change amount, and the estimated cool-flame-heat-generation-quantity-corresponding value.