Thanks for your reply. The diodes seems like a good solution. So when the wall power is plugged in, if it has a higher voltage than the battery, the wall will power the circuit. If the battery has a higher voltage (even when the wall power source is plugged in) the battery powers the circuit. If no wall power is connected, the battery powers the circuit. Is that correct?
What if the battery and wall power have the same voltage (i.e the wall adapter is plugged in and the battery starts off with a higher voltage than the wall power, but drains over time until they both have the same voltage in some instant)?
My concern is that people using this circuit will often install fresh 9v batteries, which may have equal (or greater) voltage than the 9v DC wall power supply.