It is understood Palace had a £26million bid accepted by Liverpool on transfer deadline day and the move was officially confirmed in the early hours of Friday morning.
The Eagles announced in a statement on their official website: "Crystal Palace have completed the permanent signing of Mamadou Sakho from Liverpool on a four-year deal for an undisclosed fee."
The France defender spent the second half of last season on loan at Selhurst Park having fallen out of favour with Reds manager Jurgen Klopp over a number of disciplinary issues last summer, and a return south was always the player's preferred option.
Liverpool had been holding out for £30m but after turning down £22m plus £3m in add-ons on Wednesday - Palace's third offer - the Reds decided to cut their losses on a player whom they have been trying to offload for a year.
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