Try Neem oil. It may not get rid of the scale entirely but it seems to have helped on some cuttings I got last summer.

For a more permanent solution you might have to resort to drastic measures. Take some cuttings, strip all the leaves off, spray with Neem or ultra fine oil (supposed to be good for scale), and either water root or start in sterlized soil and pot.

The other thing you might do is rinse the cut stems in a 10% bleach solution for 30 seconds or so (no idea on time or amount of damage it might do but they do this with tissue culture on begonia leaves to sterilize it completely and also do an alcohol bath on top of that - it does a lot of damage but they still have enough to start lots of new sterilized plants).

I did that (stripped the leaves and ran it through a bleach bath) with confederate rose (hibiscus mutabilis) this past year to get rid of white flies and I haven't seen a white fly on any of the new leaves.

Wow, I would think that bleach in any strength would be fatal to a delicate begonia. But hey, if I can save the plant it's worth a try. Thanks, Cheryl

I would probably recut the ends after the bath. I didn't have to recut the ends of the hibiscus but then they are woody stems.