Sandy Cooper was one of three people chosen to represent the Elgin City North ward in last week's election.
He tendered his resignation in a letter to the council's chief executive. When contacted by BBC Scotland and asked about his decision, Mr Cooper said: "It's just not for me."
The by-election will be held on 13 July, with the count the following day.
Moray Council said Mr Cooper's nomination papers contained no party or political affiliation.
A date for a by-election after a new councillor resigned only days after being elected in Moray has been set.