The pilot light will light when I hold down the igniter button but as soon as I let go the pilot light goes out.

How long are you holding down the ignitor button?

If you hold down the ignitor for 15-20 seconds and pilot goes out, it is most likely the thermocouple.

It sounds like you need a new thermocouple if you have been holding the button down for 60 seconds and the flame is on the tip for 3/8”. Thermocouples come in a kit that you use to replace the old one. It is the little rod sticking in the flame and the flame should be only on the 3/8" of the tip. Look at the one you have because there are two types: one with two wires and one that screws into the gas valve. The thermocouple produces electricity in millivolts to keep the valve open to keep pilot light on and some even send power through a t-stat and open the main gas valve, like in a floor furnace. Replace it and adjust it carefully so the flame hits just the 1/2" on tip and no more. This will fix it. They wear out every few years. If the thermocouple does not work after adjusting the flame then you have an open coil where the thermocouple screws in. It is best to check the gas valve first with a good digital multimeter set on the highest “ohms” setting. Measure either across the two terminals if you have a rare one with two wires or from the center to the gas valve body if it screws into a socket. It will have a high resistance.
Here is how to use a meter and how to test and replace thermocouples: https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20G-
20%20Electrical%20Theory%20and%20Diagnosis%20of%20 Major%20Appliances%20Service%20Manual.pdf (https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20G-20%20Electrical%20Theory%20and%20Diagnosis%20of%20 Major%20Appliances%20Service%20Manual.pdf)
https://dl.dropboxusercontent.com/u/8010163/Thermocouple_replacement.pdf
https://dl.dropboxusercontent.com/u/8010163/Thermocouple_Replacement_and_testing.pdf
https://dl.dropboxusercontent.com/u/8010163/ThermocouplesandPilots.pdf