While Mr. ionic is conjuring up the circuit from his wonderful idea, perhaps you can also consider my not so clever idea of using a simple LM317 adjustable voltage regulator.
The LM317 is a cheap three pin device working with a wide range of input voltages and allow the user to adjust its output voltage from 1.25V upwards, using two resistors or one fixed resistor plus one variable resistor.
You don't need to use a rechargeable battery but a normal 1.5V AA battery instead. The output of the LM317 should be going to the battery (+) terminal via a series 1N4001 diode to prevent the battery discharging through the voltage setting resistors should the AC power gone off.
You just need to set the LM317 voltage regulator output voltage to 2.1V(1.5V+diode 0.6V) and the AA battery will not charge or discharge under this condition and the MP3 will work happily at 1.5V. The input to the LM317 should be about 5V or more via a common AC/DC wall adaptor.
More information is available by googling "LM317".