Your current sensor runs on +/- 15 volts, right? That implies an op amp somewhere in the loop. It may have a small offset that is causing a small (12 amp) current even when the input is 0 volts. You may be able to trim this offset to zero with a small pot to the output if there is enough source resistance. Or some additional resistors may be needed.
Can you get the current control to go slightly below 0 volts? Maybe take the low arm of the control one diode drop (or less) to the - 15 volt supply?