Well, according to my trusty Guinness Book of British Hit Singles, it looks like it was Roy Orbison with 'It's Over'. On 9 July 1964, the number 1 spot went to The Animals with 'The House of the Rising Sun'.
That doesn't work. The dates are for "week ending", not "week beginning". 8 July falls within the week ending 9 July, therefore the number one was House Of The Rising Sun. And incidentally, the NME chart (which was better recognised at the time) agrees on this.