I've seen 3 ways to handle this problem:

• Dubbing the actor's lines
  This only works if the actor has at least a small degree of proficiency in the language of the movie. He or she has to be able to make plausible movements of the lips, so that a native speaker can lip-sync him or her. They did this in Arnold Schwarzeneggers debut movie, Hercules in New York.
• Sub-titling the actor's lines
  I've seen this done in a number of Scandinavian movies. One example is in the Norwegian movie "Dykket" (the Dive), in which Michael Kitchen played the role as Bricks. As Scandinavia has such a small population, and thus a shortage of good actors, they occasionally cast an actor from the English-speaking world in their native-language films. The foreign actor invariably portrays an Englishman/American/Australian/etc., who understands Norwegian, Swedish or Danish perfectly, but speaks only English. This is not as implausible as it may seem. Being able to understand a foreign language comes much earlier than being able to produce sentences in it. This applies both to the native Scandinavians, who have compulsory English classes in primary school, and to the foreigner, who has presumably been in an immersive environment since he or she immigrated.
• Not giving the actor any lines at all
  By far the most ambitious solution. I've only seen this done once, in a Swedish horror series. In one of the episodes, they cast Zelda Rubinstein (of Poltergeist fame) as the protagonist's mother. The son would say or ask something in Swedish, and the mother would intone the reply: "Mm." "Mhm?" "Mmmm!" I didn't think it worked very well. She must have had a speech defect, or maybe she was retarded.

Actually, some movies are in a language none of the actors speak. And example would be the recent Crouching Tiger, Hidden Dragon, which was in Mandarin. However, only one of the actors spoke Mandarin, so the remaining actors had to learn Mandarin. And they did. Thus, their is a 4th, very long and hard way to get around the problem.

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Update: A reader informed me that the accents the actors use are, apparently, hysterically bad. Still, it's impressive anyways...

I'm reluctant to call this the 5th way, since it's closer to the Crouching Tiger Hidden Dragon dilemma, but it's worth noting. A director can always cast the foreign character as a native mental deficent. I guess the best example of this is The City of Lost Children with New Yorker Ron Perlman, where the filmmakers had him speak in rather broken French. As an aside, it's always oddly easier for English language filmmakers to put non-English speaking actors/characters in their movies.