This injector with two valves (7, 10) which open at different fuel pressures so as to establish a primary flow regime and then a secondary flow regime is characterized in that the primary regime is enriched with fuel via an orifice (21) which adds a branch to the primary flow path of the fuel, but which is closed when the secondary valve is displaced so as to prevent enrichment of the fuel in the secondary regime. This invention applies in particular to certain aircraft engines.