Hillhead,
Its more that likely that your relative was in Holywell Hospital which is located in the outskirts of Antrim town. The records for the hospital are in PRONI and are "closed" however you could put in an FOI request for the information stating that all close relatives to William are now deceased. Also give your relationship to William and state that you wish to see them for family history purposes. It might be a shot in the dark as you don't have a death cert, but if he did die in Holywell that should be the registrar district that his death was registered.
I went through this process a few years ago and was able to get all the hospital notes on a relative who died in 1939 - he was from North Antrim and admitted to Holywell from a local hospital in Ballymoney.
Hope this makes sense.
Mags