A control apparatus for an internal combustion engine, comprising: an intake closed-valve stopping mechanism that is capable of stopping an intake valve in a closed state; an exhaust open-valve stopping mechanism that is capable of stopping an exhaust valve in an open state; and valve stopping control means that, in a case of executing a fuel-cut operation, including at a time the internal combustion engine is stopped, causes all intake valves in all cylinders to stop in a closed state by means of the intake closed-valve stopping mechanism, and causes at least one exhaust valve in all cylinders to stop in an open state by means of the exhaust open-valve stopping mechanism, wherein: the valve stopping control means comprises first valve stopping control means that stops the exhaust valves in an open state at the timing which is delayed by at least one combustion cycle or more after stopping the intake valves in a closed state, and second valve stopping control means that executes an operation that stops the intake valves in a closed state and an operation that stops the exhaust valves in an open state during a same combustion cycle; the control apparatus characterized by further comprising control switching means that determines if a fuel-cut operation is executed, and that determines whether or not a crankshaft is caused to continue idle running rotation during the fuel-cut operation, and that actuates the first valve stopping control means, when the crankshaft continues idle running during the fuel-cut operation, and actuates the second valve stopping control means, when the crank shaft stops in a case where the execution of the fuel-cut operation stops the internal combustion engine. an intake closed-valve stopping mechanism that is capable of stopping an intake valve in a closed state; an exhaust open-valve stopping mechanism that is capable of stopping an exhaust valve in an open state; and valve stopping control means that, in a case of executing a fuel-cut operation, including at a time the internal combustion engine is stopped, causes all intake valves in all cylinders to stop in a closed state by means of the intake closed-valve stopping mechanism, and causes at least one exhaust valve in all cylinders to stop in an open state by means of the exhaust open-valve stopping mechanism, wherein: the valve stopping control means comprises first valve stopping control means that stops the exhaust valves in an open state at the timing which is delayed by at least one combustion cycle or more after stopping the intake valves in a closed state, and second valve stopping control means that executes an operation that stops the intake valves in a closed state and an operation that stops the exhaust valves in an open state during a same combustion cycle; the control apparatus characterized by further comprising control switching means that determines if a fuel-cut operation is executed, and that determines whether or not a crankshaft is caused to continue idle running rotation during the fuel-cut operation, and that actuates the first valve stopping control means, when the crankshaft continues idle running during the fuel-cut operation, and actuates the second valve stopping control means, when the crank shaft stops in a case where the execution of the fuel-cut operation stops the internal combustion engine. the valve stopping control means comprises first valve stopping control means that stops the exhaust valves in an open state at the timing which is delayed by at least one combustion cycle or more after stopping the intake valves in a closed state, and second valve stopping control means that executes an operation that stops the intake valves in a closed state and an operation that stops the exhaust valves in an open state during a same combustion cycle; the control apparatus characterized by further comprising control switching means that determines if a fuel-cut operation is executed, and that determines whether or not a crankshaft is caused to continue idle running rotation during the fuel-cut operation, and that actuates the first valve stopping control means, when the crankshaft continues idle running during the fuel-cut operation, and actuates the second valve stopping control means, when the crank shaft stops in a case where the execution of the fuel-cut operation stops the internal combustion engine.