Plant for desalinating water of a 
water supply system, which comprises one or more tanks for accumulating water (2) in an immiscible manner, in order to store a softened supply thereof, provided with a first inlet connection (3) and with a first outlet connection (4) respectively connected to a feeding 
pipe (5) connected to the 
water supply system (50) and to an extraction 
pipe (6) for supplying users. The 
plant (1) also comprises a filtering unit (10) for 
water softening, for example obtained with a flow-through condenser (10") or with a 
reverse osmosis membrane filter (10'), connected in parallel to the tank (2) with second inlet connection (1 1) and second outlet connection (12) respectively hydraulically connected to the first inlet connection (3) and to the first outlet connection (4) of the tank (2). Circulating means (13) are provided which can be activated to force at least one 
water flow to be treated to pass through the filtering unit (10), producing a filtered 
water flow, which is progressively stored in the tank (2) according to a filling direction (VI) thereof. Operatively, the feeding 
pipe (5) of the 
water supply system (50) forces, when the user requests water, a 
water flow intended for use to flow through the tank (2), causing the at least partial evacuation of the filtered 
water volume that was stored therein in an immiscible manner, in an evacuation direction (V2) opposite the filling direction (VI) with which the filtered water flow had been previously stored in the tank 2.