Sight unseen I would guess that the thermocouple is the problem.
On the gas control you have a "RED" button that must be manually depressed while lighting the pilot and held down for approximately 60 seconds after the pilot is lit.
Inside the gas control there is a spring loaded normally closed shutoff valve connected to an electrical solenoid. When you depress the Red button you are manually holding the gas valve open to permit the pilot burner to light.
Once the pilot is lit, the flame of the pilot heats the tip of the thermocouple, which is comprised of a bimetal element that will generate a slight electrical current when heated. That electrical current then energizes the solenoid coil and holds the gas valve in the open position. If the pilot flame should go out, the current stops and the solenoid valve instantly returns to its normally closed position, turning the gas off. This is how the gas control can prove the presence of a pilot flame before allowing the thermostat to turn the main gas on to the burner.
There are a number of factors that can render the thermocouple inoperative.
1. The gas line pressure may have changed, resulting in lowering the height of the pilot flame to a point where it is not adequately heating the thermocouple.
2. The thermocouple may have somehow been bumped out of alignment so the tip is no longer properly centered in the pilot flame.
3. There could be scale, carbon or other debris built up on the thermocouple probe that would act as an insulator, preventing the thermocouple from being heated to the required temp.
4. There could be corrosion on the thermocouple connection to the gas valve that is causing sufficient electrical resistance to prevent the thermocouple from holding the gas valve open.
5. The most common problem is a burned out thermocouple.
Other conditions that could result in your problem is improper venting, which can cause backdrafting when the burner shuts off, and the backdrafting can be blowing the pilot flame out.
Although very rare, the problem could also be resulting from moisture droplets in the gas supply which are momentarily restricting the gas flow. It is for this reason that the gas lines are required to have "Drip Stubs".