Yep, we had this problem with an inline thermostat in a Mitsubishi FGC20 some years back. That was the first time I had ever encountered an inline thermostat and at first glance I interpreted it to be a "check valve" due to the directional arrow on the cylindrical body. It is crucial that the inline thermostat be placed in the RETURN circuit hose so it can sense the coolant temperature LEAVING the regulator with the arrow on the thermostat pointing away from the regulator.

ONE more thing to check as far as coolant flow thru the regulator is to look and see if it has an inline thermostat in one of the coolant lines for the regulator, if this is defective then your flow will be greatly decreased, this thermostat is cylindrical in shape and looks more like an AC fuse when you look at

L1ftmech would be correct on this one
flow is the key , if you have poor flow of coolant going through the vaporizer then sure it will freeze up. Even if the coolant level is ok in the radiator if the flow is poor it will still freeze up.

I would check the flow in the radiator first, if you dont have coolant noticably moving around inside the radiator then it could be a waterpump issue, also the bypass circuit could be clogged up via a hose or a port at a fitting. These things will require you to drain the coolant and start pulling hoses and checking fitting orfices etc.
example: i had a cat gp25k with the mitsu 4G64 engine doing this same exact thing, i checked everything and could not find the cause for poor circulation. Only when i drained the coolant and started checking hoses and fitting orfices i found a buildup of silicates inside the intake where the coolant circulates through it, it had built up so bad it had clogged the intake up therefore not letting the coolant circulate through the bypass circuit. Yes the bypass circuit in the cooling system is the part the LP system uses to get its coolant from.

I've even seen in a worse case scenario where the block was clogged up and the head had to be pulled off and the ports cleaned out.

I know this sounds like a bit much to do but if you cannot find the obvious then you have to think outside the box and start digging deeper. There is always a cause of a symptom, you just have to figure it out and find it ;o)

Good luck

Bad diaphrams -replace with overhaul kit or whole converter. Should be a Impco Model J or K ( Cobra).

Giving a generalized reply here.
The first thing to check is the level of coolant in the radiator.
Coolant level falls too low and there may not be sufficient volume of coolant flow through the regulator.
Even though you may have coolant "present" in the regulator, it is imperative that there be sufficient flow of coolant through the regulator to heat it up enough to stay ahead of the cooling effect of the evaporating liquid propane.
I would first make certain that neither of the coolant hoses serving the regulator are kinked nor swollen (swelling may indicate a deteriorated inner lining of the hose resulting in a "flapper type" restriction caused by the hose itself).
Icing that occurs after about 20 minutes is often the result of poor coolant flow rather than overfueling by a defective regulator. You want to test the temperature of both coolant hoses serving the regulator. If you have good coolant flow......both hoses should be as warm as the top radiator hose after 20 minutes of operation. If one hose is cold and the other hose only slightly warm when the regulator begins to ice up, you have a coolant flow issue.
If you can eliminate poor coolant flow then the regulator is certainly processing too much liquid propane.