The other answer's implicit suggestion that there is reasonable doubt that Dumbledore did not know Sirius was innocent is not quite compatible with the book. Specifically, near the end Dumbledore stated why Sirius had been imprisoned and why he now believed Sirius to be innocent:
[...] Dumbledore turned to Harry and Hermione. They both burst into speech at the same time.
But Dumbledore held up his hand to stem the flood of explanations.
"It is your turn to listen, and I beg you will not interrupt me, because there is very little time," he said quietly. "There is not a shred of proof to support Black's story, except your word -- and the word of two thirteen-year-old wizards will not convince anybody. A street full of eyewitnesses swore they saw Sirius murder Pettigrew. I myself gave evidence to the Ministry that Sirius had been the Potters' Secret-Keeper."
"Professor Lupin can tell you --" Harry said, unable to stop himself
"Professor Lupin is currently deep in the forest, unable to tell anyone anything. By the time he is human again, it will be too late, Sirius will be worse than dead. I might add that werewolves are so mistrusted by most of our kind that his support will count for very little and the fact that he and Sirius are old friends --"
"Listen to me, Harry. It is too late, you understand me? You must see that Professor Snape's version of events is far more convincing than yours."
"He hates Sirius," Hermione said desperately. "All because of some stupid trick Sirius played on him --"
"Sirius has not acted like an innocent man. The attack on the Fat Lady -- entering Gryffindor Tower with a knife -- without Pettigrew, alive or dead, we have no chance of overturning Sirius's sentence."
"But you believe us. "
"Yes, I do," said Dumbledore quietly. "But I have no power to make other men see the truth, or to overrule the Minister of Magic...."
"What we need," said Dumbledore slowly, and his light blue eyes moved from Harry to Hermione, "is more time. "
"But --" Hermione began. And then her eyes became very round. "OH!"
"Now, pay attention," said Dumbledore, speaking very low, and very clearly. "Sirius is locked in Professor Flitwick's office on the seventh floor. Thirteenth window from the right of the West Tower. If all goes well, you will be able to save more than one innocent life tonight. But remember this, both of you: you must not be seen. Miss Granger, you know the law -- you know what is at stake... You -- must -- not -- be --seen. "
According to Dumbledore, he had believed Sirius to be guilty because he had believed him to be the secret keeper for the Potters, not knowing that they had changed it to Peter at the last minute. Earlier, Sirius himself had told Harry that he considered himself guilty for persuading the Potters to change the secret keeper:
"Harry... I as good as killed them," [Sirius] croaked. "I persuaded Lily and James to change to Peter at the last moment, persuaded them to use him as Secret-Keeper instead of me... I'm to blame, I know it... The night they died, I'd arranged to check on Peter, make sure he was still safe, but when I arrived at his hiding place, he'd gone. Yet there was no sign of a struggle. It didn't feel right. I was scared. I set out for your parents' house straight away. And when I saw their house, destroyed, and their bodies... I realized what Peter must've done... what I'd done.... "
And that is why he did not even attempt to exonerate himself when sentenced to Azkaban, because he felt guilty even though he was not. It is believable that he never told Dumbledore, for the same reason, because at that time it was not known that Voldemort survived, so there was no real need to expose Peter who had fled.
Also, it is likely that Sirius had his wand pointed at Peter when Peter cut off his thumb behind his back and then blew up the street behind him before vanishing into the sewers below. The eyewitnesses would have seen Peter gone after the explosion, and conclude that Sirius was responsible.