In the past 10 drafts (2002-2011), these are the numbers of QBs drafted in the 1st round by team. Bolded teams are currently settled at the QB position because of that draft pick.
Arizona Cardinals: 1
Atlanta Falcons: 1
Baltimore Ravens: 2
Buffalo Bills: 1
Carolina Panthers: 1
Chicago Bears: 1
Cincinnati Bengals: 1
Cleveland Browns: 1
Dallas Cowboys: 0
Denver Broncos: 2
Detroit Lions: 2
Green Bay Packers: 1
Houston Texans: 1
Indianapolis Colts: 0
Jacksonville Jaguars: 2
Kansas City Chiefs: 0
Miami Dolphins: 0
Minnesota Vikings: 1
New England Patriots: 0
New Orleans Saints: 0
New York Giants: 1
New York Jets: 1
Oakland Raiders: 1
Philadelphia Eagles: 0
Pittsburgh Steelers: 1
San Diego Chargers: 1
San Francisco 49ers: 1
Seattle Seahawks: 0
St. Louis Rams: 1
Tampa Bay Buccaneers: 1
Tennessee Titans: 1
Washington Redskins: 2
17 of the 29 QBs (59%) taken in the 1st round during that period have not achieved the intended result, i.e., an adequate, long-term solution to the position.
Only 12 of the 29 QBs (41%) drafted in the 1st round during that period have resulted in an adequate, long-term solution to the position.
The odds are against solving the quarterback position, even with a 1st-round pick.
Now, there may not be a better way of doing it, but doing it even in the 1st round stands you a better chance of being unsuccessful than successful.