Diana Spencer, was the only daughter and second child of Hon. John Spencer, of Althorp, and his wife, Lady Georgiana Carteret.
Her dates of birth and death vary in the sources. A birth date of May 1735 is given in Collins, Peerage of England, and followed by other sources, but this is impossible as her brother John Spencer, later 1st Earl Spencer, was born on 19 December 1734. She was baptised on 11 June 1736, and although her birth date on the register isn't totally clear, it is possibly 17 May 1736 (though it has been transcribed as 17 April that year).
Although Debrett's peerage lists her death as occurring in 1771, it seems more likely she died as a child. Collins, Peerage, states that she died "about 6 years old", and the Wikipedia article for her father, that she died aged 8, three years before her father, who died 20 June 1746, but there aren't any sources for this statement. There is also a FindAGrave record for her, which states that she died 12 June 1743, and was buried the following day in the churchyard of the Church of St Mary the Virgin and St John, Great Brington, Northamptonshire, England. Although this also doesn't cite any sources and hasn't been confirmed, it seems the most likely at this point.
Cokayne, George Edward, The Complete Peerage: or a history of the House of Lords and all its members from the earliest times, 2nd ed., vol. 12 (part 1), rev. & ed. by Geoffrey H. White, London: St Catherine Press, 1953. Electronically produced by ABC Publications, London, 2000.
St George, Hanover Square (Westminster, Middlesex, England), Baptisms & Burials (date range unknown), p. 289, "Diana Spencer of the Hon. John Esqr & Georgina" baptised 11 June 1736; digital images, Findmypast.co.uk (http://search.findmypast.co.uk : viewed 6 June 2015), from Westminster Archives.