I was getting increasingly frustrated but knew there must be a way round this. Many of the pages I found didn't include pictures so it took me a while to find exactly where to resolve this problem.
Let's do this!
So, to export an Illustrator file to a JPEG, go to File, Export...
The export dialogue box will open
Then, when Illustrator prompts you to browse to a folder you want to save your JPEG in, check the Use Artboards checkbox on the bottom left of the dialogue box.
This will crop the artwork to the artboard with no need for a clipping mask. Hurray!
|Exported JPEG using Use Artboards checkbox|