I have the same issue with my Peugeot Sum-Up 125 which I bought in Oct 2008
Was fine for the first year (which is all my warranty covered), but this winter it simply simply stopped working in cold/damp weather.
It is stored in a Garage when not in use and only works in cold weather if I have left it in there for a few days. It will then work for a week and then cut-out again until I leave it in the Garage for 3 - 4 more days.
I have done the following:-
  * Had the Throttle Cable replaced (was recalled for this)
  * Bought and changed the Spark Plug
  * Recharged the battery.
  * Changed the air filter (in case this was expiring and getting damp was preventing air getting to the engine)
  * Adjusted the idle speed up/down (no difference)
In the posts below it mentions "Carb Freezing Up" which I cant fix under Warranty. Has anyone any more details on this?
                
 Gareth
                
 February 2010