I should just like to mention how wonderfully helpful the people of the Northern Territory are, as are the kind people here at RootsChat. I spoke to the registrar of BMD in Darwin, who kindly read out the certificate to me (I am purchasing it, too) and she then gave me a telephone number for the people who look after the cemetery, who found the records for my great-grandfather and offered to take a photograph and send it to me.
I also spoke to someone at the supreme court who is going to try and locate the records for me. And, thanks to craizi daizi, I also found the notice from the public trustee asking for beneficiaries, creditors, etc. to come forward, which was not published nationally, meaning neither my grandfather nor his sister would have been aware of my great grandfather's death or that his estate was in the hands of the public trustee.