An exhaust-gas recirculation apparatus for an internal combustion engine has a high-pressure EGR passage (20), a low-pressure EGR passage (21), a high-pressure EGR valve (22), and a low-pressure EGR valve (24) and performs open-loop control to the low-pressure EGR valve (24). The flow rate (Ghpl) of EGR gas flowing in the high-pressure EGR passage (20) and the flow rate (Glpl) of EGR gas flowing in the low-pressure EGR gas (21) are estimated based on the oxygen concentration (O2s) in the gas at the portion of the intake passage (3) downstream of the portion to which exhaust gas is delivered from the high-pressure EGR passage (2), the flow rate (Gdpf) of the gas passing through an exhaust-gas purifying catalyst unit (11) provided in the exhaust passage between the exhaust-gas inlets of the high-pressure EGR passage (20) and the low-pressure EGR passage (21), the flow rate (Gafm) of fresh air, and the fuel amount (Q) supplied to the cylinders (2) of the internal combustion engine per unit time.