LA Galaxy have struggled to make an impact in the MLS this season, and are outside the play-off places in the Western Conference.
"I realise we have a lot of work ahead of us and I am optimistic that with the support of management, our coaching staff and the players, we will move forward," he added.
I am impressed with Bruce's passion, desire and focus
Tim Leiweke, president of Galaxy owners AEG
Arena has been out of work since parting company with MLS side New York Red Bulls in November last year.
Tim Leiweke, president of Galaxy owners AEG, added: "Bruce is the best coach for the Galaxy and certainly one of the best coaches in the history of the league.
"While it has been a very difficult week, I believe the end result is a very positive step forward for the Galaxy.
"I am impressed with his passion, desire and focus and believe that his recent experiences with the US national team and Red Bulls have prepared and motivated him for this next challenge."
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