Install a fuel shut-off valve up close to the fuel tank outlet, and use it when you shut the engine off when done using it! That may cure the problem. RJ
PS: I'm assuming it doesn't have a fuel pump.
PPS: If it still leaks out, then a carb fix may be in order.

That's a Briggs 42E707-2275-E1

Fuel pump is part of the carb.
Probably a check valve in the pump is leaking or the fuel line connection to the pump has an air leak.
Carb bowl SHOULD retain enough fuel to start. It must be seeping out during non use too?

FP repair kit would likely fix one problem.

I'm with Bill on this one.
The fuel bowl contains the fuel that supplies the initial start up.
The engine should run for perhaps as long as a minute (longer at idle RPM) just on the fuel contained in the bowl.
If you are having to crank the engine for an excessive length of time before it begins to run, make sure the choke plate is shutting completely when choke is applied.
It requires a very rich (in fuel) mix to get a cold engine going.
The choke does that job.
Check choke operation first.

Thank you for all your replies. I'm going to go with the check valve. I've already replaced the fuel line, filter and cleaned carb. Would the check valve cause the engine also to surge at idle?

No, the check valve is a part of the fuel pump, really just a flapper type valve on these. It will not cause surging at idle.
What about the choke, is it closing all the way when applied?
Surging at idle on these can sometimes simply be due to the idle speed being set TOO low. Realistically speaking, a partially clogged idle or pilot jet could cause surging at low end RPM.
A float bowl set too low may also cause some surging at low RPM.
Your surging issue might be some combination of both those details.

The choke fully closes when it's on. It actually starts fine when the engine is cold but when it gets hot it tends to run lean then shudder to a stop. It won't start again until it cools down.

Hmmmm.
After reading the latest description of the problem, I suggest you begin looking for a cause of engine overheating.
If you have never pulled the sheet metal shrouding off the engine, there may be a blockage of accumulated grass clippings, or worse..........mice activity.
Mice LOVE to build nests and/or food caches inside the cooling ducts formed by the shroud panels.
Engines have been destroyed when the nests were big enough to block all air flow.
If the blockage is not so great as to stop ALL air, it could still result in the engine getting hot enough to lean out/vapor lock.

Any real diagnosis of this issue would include testing for loss of spark when the engine "shudders to a halt".
It seems that these boxer twins have an Achilles heel in their Magnetron ignition that is very sensitive to "heat soak" after they have been in service a long time.
But usually, if it is due to a heat soaked Magnetron, the real complaint is that the engine will not RESTART after a hot shutdown has been done deliberately (such as the operator needing to refill the tank or get a drink of water).
Because the Magnetron unit sits right next to the flywheel, it is washed in fresh cooling air from the fan on the flywheel and can't really heat up until the engine is shut down and radiant/convective heating begins from the hot engine below.
I am pretty sure at this point that the issue you thought you saw about the fuel emptying is nothing but a red herring dragged across the trail.
Is your fuel filter a "transparent plastic" unit?
If so, don't ever look at it again in hot conditions. It is perfectly normal for a fuel filter to APPEAR empty at times under hot conditions and has been the cause of many erroneous diagnoses because it was interpreted as a defective condition.

Thanks Mownie, I did have mice living in the engine fins and I thought I had cleaned out everything thoroughly but after you mentioned it I went back and did a more thorough job and found places between the fins the size of at least two screw driver widths. ( which was my tool of choice to do the cleaning). The engine runs much better and does not seem to over heat. Although it does not want to start once it is switched off. Could the magnetron be easily replaced to alleviate that problem?

I would not go so far as to tell you "changing the Magnetron will eliminate the problem" even though I think it might.
Evasive answer? Not so much that as I just would want to be absolutely certain that the engine was really losing all spark when it does not want to restart.
And as has been pointed out in numerous threads (not this one), the only accurate way to really determine whether an ignition is REALLY making a spark (or not) is to test the high tension circuit while it is attached to the spark plug with the spark plug actually in place in the engine.
To do this requires a spark tester that can be attached, and viewed by the testing person when the engine is running..........or cranking.
The reason that simply using a spare spark plug and laying it on the engine where the gap can be seen is that an ignition system may throw a spark under atmospheric pressure but be unable to make a spark when the spark plug is inside the combustion chamber under compression pressure and in the presence of gasoline.
If I KNEW that the ignition was unable to throw a spark when hot, then of course I could recommend replacing that Magnetron, but as they are sorta pricey, I don't want you to spend the money unless it is proven to be defective.
Now, there is another thing you might do to see if this reluctance to restart when hot is carburetor related.
You can pour a 1/2 gallon or so of cool water on the exterior surfaces of the carburetor and the intake manifold when it does not want to restart.
This will reduce the temperature of the parts so the air/fuel mix entering the engine will not be so LEAN as to not support combustion. If this helps the engine restart, it implies that the engine may still be running a little warm.
Also, there actually is a "hot starting" trick that I and some other members here have used with some success.
This trick is to move the throttle to the wide open position, apply the choke fully but keep your finger on the choke control and as soon as you turn the key to START and the engine moves.............push the choke completely off.
If none of these works then you may want to get a Briggs spark tester and check that Magnetron under real life conditions.

This post was edited by mownie on Sat, Jul 13, 13 at 22:58