For everyday purposes we can say that yes, the same current will flow through the circuit. If we look a little deeper into it we would see that it depends on the internal resistance of the battery and how much current your power adapter could provide at 12 volts. For example, a 12 volt power adapter plugged into a circuit with only 1 ohm of resistance would be expected to put out 12 amps of current. The power adapter may or may not be able to supply 12 amps of current, as it may be current limited.

http://en.wikipedia.org/wiki/Internal_resistance
http://en.wikipedia.org/wiki/Current_source
http://en.wikipedia.org/wiki/Current_limiting
http://data.energizer.com/PDFs/BatteryIR.pdf