Use AC-DC adapter instead of AC-AC adapter
First of all the 7812 (positive voltage) and 7912 (negative voltage) regulators need about 2v higher input than the output in order to operate properly (that is at least 14v for the 12v regulator and -14v for the -12v regulator) .

The AC voltage in the input is rectified by two diodes to generate a symmetric unregulated supply (that is a positive and a negative voltage with  reference to the ground) which feeds the regulators inputs and you get +12v and -12v regulated rails at the output. If you feed the input with 12v DC there is no way for the circuit to generate the negative rail voltage so this can't work.

Here is a link that explains the operation of the diodes in the circuit you show
http://metroamp.com/wiki/index.php/Half_Wave_Voltage_Doubler






  I've got a 12V AC-DC adapter that delivers 200mA


Depending on the internal implementation of the AC-DC adapter it may be possible to use it.
If it is the classic transformer type rather than a switching type (you can easily judge it by size and weight) then you can omit the output circuitry (you'll need to open the adapter case to do that) and connect the AC output of the transformer to the AC input of the O2 power supply circuit.