In a device for driving LEDs with variable light intensity, a supply stage has a first operating mode, in which a controlled supply current is generated, and a second operating mode, in which a controlled supply voltage is generated. A LED is connected to the supply stage, receives the controlled supply current or voltage, and has a turning-on threshold voltage higher than the controlled supply voltage. A current sensor generates a current-feedback signal that is correlated to the current flowing in the LED and is supplied to the supply stage in the first operating mode. An intensity-control stage generates a mode-control signal that is sent to the supply stage and controls sequential switching between the first and the second operating modes of the supply stage.