Did it in mid-July last year and the roads were clear and dry and the weather great. But some of the guys with whom we met up said it had been snowing heavily just a couple of days before. I think there's always a chance you'll be unlucky, it's just a case of minimising those chances.
We rode it going west to east. Going up was amazing, fantastic, tremendous fun, but coming down the eastern side was only amazing and fantastic. I wish we'd had time to do it the other way round.
Incidentally, Livigno (30-odd miles west of the pass) is a great place to stay - loads of hotels, and the lowest tax on drink and petrol of anywhere I've ever been.
While the Stelvio is special, don't forget there are hundreds of miles of brilliant roads and some equally good passes. Everyone has their own favourite -I really enjoyed the Bernina pass to the south of Livigno which was smaller (although with some stunningly fast bits too), far less busy and very scenic.
My favourite road - of the ones I've done - in the Alps is the Grossglockner in Austria - within a brilliant days ride round the Dolomites of Stelvio. (A road built by "Unemployed Austrians in the 1930's" according to the publicity as a way to avoid mentioning slave labour.)
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