The least popular French president for decades, Hollande's poll rating fell five points, dipping below 20 percent for the first time since coming to power in 2012.
In contrast Valls, who took office late last month, becomes the most popular French premier at the start of his tenure, according to the monthly findings of the Ifop polling institute which were published in the weekly Le Journal du Dimanche.
The opinion poll findings also exposed the biggest gap between the popularity of a French president and prime minister.
Socialist Hollande, who has been under scrutiny for his private life as well as his politics, recently brought back his former partner Ségolène Royal from the political wilderness to join a new, streamlined government, after his Socialist Party suffered an election drubbing.
The stinging setback in nationwide municipal elections prompted Hollande to sack prime minister Jean-Marc Ayrault and replace him with tough-talking former interior minister Valls, 51.
The 60-year-old Royal had in the past been one of the Socialists' biggest hitters, but was reportedly blocked from Hollande's first cabinet because of hostility from Valerie Trierweiler, the president's then girlfriend and de facto first lady.
That obstacle has now been removed following Hollande's split from Trierweiler in January in the aftermath of the revelation of his affair with actress Julie Gayet.
Valls faces the tall challenge of turning around the unpopular government's fortunes in time for Hollande to stand a chance of winning a second five-year term in 2017.
Valls is very clearly on the right of the governing party, having once proposed dropping "Socialist" from its name.
He is popular with voters across the political spectrum but his style and politics, often compared to those of former British premier Tony Blair, have alienated more left-leaning party members.