AC signal to switch on a DC circuit
simulate this circuit &ndash; Schematic created using CircuitLab

Here is a sample of the circuit that should do this:
The diode does the rectification, R1*C1 provide current limiting/filtering and R2 discharges the cap when AC is off. Things to keep in mind:
1) Time constant of the circuit is large enough to not alter the gate voltage substantially during the cycle of the AC
2) The rectified voltage is less than the max Vgs of the transistor but still allows M1 to fully turn on
3) Depending on the load, you may want to implement the circuit as a high or low side switch (see dashed boxes). 
4) You on time after AC is off will be determined by C1R2 time constant and the turn off voltage (Vgs) of M1