President Milos Zeman told Sobotka that he looked forward to a "strong and successful government for the Czech Republic" at a ceremony at his Prague Castle headquarters.
The two men fell out in 2003, but Zeman was forced to name his rival as prime minister after Sobotka struck a three-party coalition deal earlier this month.
Sobotka's coalition, which includes the billionaire-led populist ANO party and the smaller centrist Christian Democrats, commands a majority of 111 seats in the 200-member Czech lower house of parliament.
The Czech Republic has been mired in a political crisis since June, when then prime minister Petr Necas stepped down after his chief-of-staff and lover Jana Nagyova was arrested and charged with bribery and abuse of power.
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Prosecutors believe she had military spies tail Necas's wife. Necas has since divorced his wife and married Nagyova.
A snap election was subsequently called for October and won by Sobotka's party.
But a non-partisan cabinet appointed by Zeman in July had continued to lead the country while political leaders battled to secure a coalition deal.
Jan Outly, a political analyst at the Metropolitan University in Prague, said he did not expect Zeman to show much resistance to Sobotka's cabinet line-up.
"Of course Zeman will try to have his way, but he won't go as far as to compromise his appointment (of Sobotka) because of his dislike of some names," he told AFP.
Sobotka himself said he had heard no "negative signals" from Zeman.
The two men fell out over the new premier's reluctance to back Zeman in his bid for presidency in 2003.
Babis, who heads the ANO party, is tipped to become finance minister despite allegations that he collaborated with the communist-era secret police and has potential conflicts of interest due to his vast business empire.