The deal had been waiting for the verdict of the US Federal Trade Commission and Department of Justice following its scrutiny into the acquisition.
Delays in dealing with this scrutiny meant the takeover had to be postponed from 7 January to 22 January and then from 22 to 5 January, as Sapient will pass into "foreign" hands.
Publicis expects to complete the purchase after 11.59 EST (5pm GMT) tomorrow.
The company has "received all antitrust and regulatory approvals in connection with its tender offer to purchase, so the takeover can proceed," according to a Publicis statement.
Publicis will buy Sapient for $25 per share, or a total of $3.7 billion (its shares are trading on Nasdaq today at $24.87 giving the company a market capitalisation of $3.5 billion). When the deal was announced in November 2014, the price represented a 44 per cent premium over the share price.
Sapient is an important strategic investment for Publicis Groupe, following the collapse of the proposed deal between Omnicom and Publicis amid cultural differences.
Sapient, which works as a digital ad agency, a digital consultant and data management agency, was described as a "crown jewel, a one of a kind company born in the technology space" by Maurice Lévy, the chairman and chief executive of Publicis Groupe.
Publicis hoped the acquisition combined with its digitally focused agencies like Razorfish and DigitasLBi will help push overall digital revenues above 50 per cent.