He will remain in charge of the Baggies for their last two games of the Premier League season.
The 64-year-old will lead his country after managing 18 teams, including three national sides, during a coaching career spanning 36 years.
"Let's hope we can get everybody behind us and make certain that we work as a team and help the team get the results everyone in England expects and wants," said Hodgson.
Hodgson's contract at West Brom runs out on 30 June but the Premier League club said they have agreed to release him after their final game of the season on 13 May so he can prepare for Euro 2012.
England have two friendlies, at Norway on 26 May and home to Belgium on 2 June, before their first match in the tournament in Poland and Ukraine against France on 11 June.
The Football Association said in a statement it was "delighted" to announce the appointment of Hodgson, whose contract also covers the 2014 World Cup and Euro 2016.
The men who managed England
"Hodgson has won a total of eight league titles in a distinguished career, and coached the national teams of Switzerland, Finland and the United Arab Emirates," it added.
"Along with his vast experience of international and European football, Hodgson is the only English manager currently working in the top flight to have won the League Managers' Association Manager of the Year award."
The FA announced on Sunday it had approached Hodgson for the job despite widepsread speculation linking Tottenham boss Harry Redknapp with the vacancy after Capello quit in February.
Hodgson guided Switzerland to the 1994 World Cup and the side achieved a Fifa ranking of third in the world as they qualified for Euro 1996.
His former clubs include Inter Milan, Blackburn, Grasshoppers, FC Copenhagen, Fulham and Liverpool. He has also managed in Norway and Sweden, and been a member of Fifa and Uefa's technical study groups at tournaments.
Hodgson's Premier League record
Goals per game
Goals conceded/ game
Hodgson leaves West Brom after 15 months at the club.
Chairman Jeremy Peace said: "We did not want Roy to leave and he will be a tough act to follow.
"But we understand Roy's desire to take the England job and he will leave us after the Arsenal game on 13 May with our best wishes.
"Of course, with Euro 2012 around the corner, we appreciate that some of Roy's focus over the next 12 days will be on England matters. But we are delighted he will be seeing the season out with us.
"The board will now concentrate on the important task of appointing a new head coach to continue our project as we prepare for a third successive season in the Premier League."
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