One hundred days after an oil well operated by BP ruptured in the Gulf of Mexico, and 13 days after crews finished capping the well to contain the gushing crude, the company's incoming Chief Executive Officer Bob Dudley says the worst may be over.

"I think -- no guarantees -- but I believe there will be no more oil flowing into the Gulf as of the 15th of July," Dudley told CNN's "American Morning" on Wednesday.

No anomalies or breaches have been detected at the well, and pressure is rising slowly, indicating that it is structurally sound and has integrity, BP said Wednesday.

Dudley, whom BP named Tuesday to replace CEO Tony Hayward on October 1, said that resolving the crisis is "the single highest priority for BP going forward."

"The only way you can build a reputation is not just by words but by action," Dudley said. "I picked up that people think that, well, once we cap this well, we're somehow going to pack up and disappear. That is certainly not the case. We're -- we've got a lot of clean-up to do. We've got claims facilities. We've got 35 of those around the Gulf coast. As of this morning, we wrote a quarter of a billion dollars in checks, for claims. There's still more to go. We know that. We haven't been perfect at this. But it's a deep, deep personal commitment from me for BP and the many people in the Gulf coast to make this right in America."

BP estimates that in August, it will pay at least $60 million in advance to Gulf coast claimants who have lost income or net profit because of the oil spill in the Gulf of Mexico, the company said in a statement Wednesday.

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