Any distance that the controller thinks that he can safely get the departure airborne, as a general rule of thumb, where I work, if the arrival is outside 3.5 miles (depending on wind conditions, weather, departing aircraft type {not a heavy} ) then it is possible to depart an aircraft ahead.
If the gap is tight then I am using reduced separation in the vicinity of an aerodrome. Some units have deemed separations between arrivals and departures in tight gaps (providing the departing aircraft is xxxx metres along the runway when the arrival reaches the threshold then they are deemed separated and a "land after the departing" clearance can be issued)