In December, she announced she was stepping down as the broadcaster’s political editor after seven years.
Kuenssberg will start her role when the show relaunches this September with a new set, title, format and title music.
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She said: “I couldn’t be more delighted. For decades Sunday morning has been the moment to explore the events that shape us and to challenge and listen to our politicians.
“It’s an honour to take the chair for that conversation in the 2020s.”
Kuenssberg will take over from Sophie Raworth, who has been presenting the show, currently titled Sunday Morning, since January, following the departure of Andrew Marr.
Interim director of BBC News Jonathan Munro said: “Laura’s the perfect host for our flagship weekend politics show – she’s an engaging presenter and a razor-sharp political interviewer, and she knows exactly which questions audiences want answered.”
BBC chief content officer Charlotte Moore added: “Laura’s one of the BBC’s biggest talents and I’m delighted she’s becoming the new face of Sunday mornings.
“I’m very much looking forward to seeing how she makes the show her own.”