A stop control apparatus to be applied to an engine which includes a cylinder , an output shaft configured to rotate by receiving energy of combustion in the cylinder , an intake passage through which intake air introduced into the cylinder flows, an exhaust passage through which exhaust gas exhausted from the cylinder flows, and an EGR passage which connects the intake passage and the exhaust passage together, the stop control apparatus comprising: an injector configured to supply fuel to the cylinder ; an EGR valve which is provided to the EGR passage in an openable and/or closable manner; a throttle valve which is provided in an openable and/or closable manner to the intake passage on an upstream side of a connection portion between the EGR passage and the intake passage ; a determination unit configured to determine whether or not an automatic stop condition for automatically stopping the engine is satisfied; and a control unit configured to control the injector , the throttle valve , and the EGR valve , wherein in a case where the determination unit confirms that the automatic stop condition is satisfied, the control unit is configured to execute fuel cut to stop supply of fuel by the injector and execute oxygen concentration adjustment control which controls the throttle valve and the EGR valve such that an oxygen concentration in the intake passage on a downstream side of the connection portion becomes a predetermined target value (Dx) in a period until rotation of the output shaft stops after the fuel cut is executed, and the oxygen concentration adjustment control includes first control which sets the throttle valve and the EGR valve to a valve-open state such that the oxygen concentration is elevated, characterized in that the oxygen concentration adjustment control further includes a second control which closes the throttle valve and increases an opening of the EGR valve at a time point (t3) when the oxygen concentration is elevated to the target value (Dx) such that an intake air pressure does not largely lower and its value is maintained at a value which does not fall below a reference pressure (Px). an injector configured to supply fuel to the cylinder ; an EGR valve which is provided to the EGR passage in an openable and/or closable manner; a throttle valve which is provided in an openable and/or closable manner to the intake passage on an upstream side of a connection portion between the EGR passage and the intake passage ; a determination unit configured to determine whether or not an automatic stop condition for automatically stopping the engine is satisfied; and a control unit configured to control the injector , the throttle valve , and the EGR valve , wherein in a case where the determination unit confirms that the automatic stop condition is satisfied, the control unit is configured to execute fuel cut to stop supply of fuel by the injector and execute oxygen concentration adjustment control which controls the throttle valve and the EGR valve such that an oxygen concentration in the intake passage on a downstream side of the connection portion becomes a predetermined target value (Dx) in a period until rotation of the output shaft stops after the fuel cut is executed, and the oxygen concentration adjustment control includes first control which sets the throttle valve and the EGR valve to a valve-open state such that the oxygen concentration is elevated, characterized in that the oxygen concentration adjustment control further includes a second control which closes the throttle valve and increases an opening of the EGR valve at a time point (t3) when the oxygen concentration is elevated to the target value (Dx) such that an intake air pressure does not largely lower and its value is maintained at a value which does not fall below a reference pressure (Px). <EOS>