Thanks to Heather for her comment on my previous post. For her, and for any other Jane Austen fans thinking of visiting Hampshire - there are other Jane sites near Chawton.
The house in Winchester where she died can be seen, although it isn't open to the public - but, more importantly, she is buried in Winchester Cathedral.
Unfortunately the Cathedral charges admission. However you can also admire the Anthony Gormley statue in the crypt - which is one of my favourite pieces of art - and an altar cloth by Alice Kettle. There is also a restaurant in the Cathedral close which is open all year!