Dispatch can be spelt dis- or des-. The OED has a note that the almost universally preferred spelling until the early 19th century was 'dispatch'. Samuel Johnson did not include 'dispatch' as an entry in his great dictionary of 1755 (the headword Dispassionate is followed by Dispel), but did include 'despatch'. It was because of this that the 'despatch' spelling became common later. The influential grammarian of the early 19th century, Lindley Murray, regarded Johnson's dictionary as the arbiter when it came to disputed spellings and advised teachers to correct schoolboys' spelling of 'dispatch' to 'despatch'.
James Murray, the original editor of the OED, called the inclusion of 'despatch' an error and said that Johnson himself always used the spelling 'dispatch'. Indeed, one of the citations at the 'dispatch' entry in the OED, from Johnson's periodical The Rambler of 1751 uses the spelling 'dispatch' ("I was in my eighteenth year dispatched to the university").
More recently, scholars have found instances in Johnson's letters and other works where he sometimes spells the word 'despatch', so its inclusion in the dictionary cannot have been merely an error, as Murray said.
Johnson says that 'despatch' comes from the French dépêcher, and that is presumably why he spelt it with an e, but the OED says that, in fact, it comes from the Italian dispacciare.