Hmm, well, it doesn't look 'dappled' any more to me. This plant's real name is Salix Hakuro Nishiki, and it should start the year with leaves which are pale green, cream and tinged with pink, image here:-

http://www.ornamental-trees.co.uk/salix-integra-hakuro-nishiki-tree-pp340

Later in the season, the leaves pale down to just cream and green, but I'm not seeing any of that on this plant, the leaves all seem to be plain green, not to mention larger than they should be. Now that might be just a trick of the light in the photo, but is any of this plant variegated in any way at all?

I'm just wondering if a different rootstock has been used and has taken over quite a lot of the plant, which would explain all the green leaves... and maybe the bit that's died back now was the remainder of the original variety....

The link you sent with a picture is exactly what my willow looked like early in the season. It was beautiful! The past few weeks it has turned somewhat all green, but yes, it does still have that dappled look on some of the branches. I love this plant and want to keep it, but now I'm worried. I am going to trim it a bit and see what happens. The branches that have the dried bits are still soft and green underneath (I did a little scratching) so I'm hopeful. I would take a cutting into a local garden centre here but from experience I find the employees pretty useless when it comes to diagnosing problems. Maybe I'll have to wait a whole year to find out if it's going to survive or not. Thanks for your input!

Well there's no obvious disease apart from this die back, not even at the base of the plant, and the cause of that could be environmental, i.e., its been caught by a weedkiller spray or those parts got frazzled by hot sun, if you had any. So fingers crossed that's all it is...

Willow roots very easily so you could take a cutting as insurance in case the trouble spreads.