99.99998754%.
Actually, I don't know. The number above has been determined by both my credence in the atomic theory and the shape of my keyboard, anchoring and perhaps many other biases; the biases could have driven the number far away from its true value. But I dislike dodging questions about probability and believe that disagreements would be easier to resolve if people shared their probabilities when asked, so I provide an answer which is as good as I can make it.
I believe that there is a useful reformulation of similar questions: how many bits of evidence against the atomic theory would make you to doubt the theory on the level of approximately 50% confidence? People are much better at evaluationg the strength of their beliefs when they are an about that level, so this question could be in principle settled experimentally. Of course, people aren't ideal Bayesians and the persuasive power of evidence would depend on the way it is presented, perhaps a sufficiently clever demagogue can talk me out of my belief in the atomic theory without any evidence at all, so the approach has its limits. But it still seems that "how much evidence would it take to change my opinion" is an easier question to tackle directly than "what's the probability that I am right". Unfortunately, I can't answer that for the case of the atomic theory without some hard thinking, and I am lazy to think about it hard just now.
How much confidence do you place in the scientific theory that ordinary matter is made of discrete units, or 'atoms', as opposed to being infinitely divisible?
More than 50%? 90%? 99%? 99.9%? 99.99%? 99.999%? More? If so, how much more? (If describing your answer in percentages is cumbersome, then feel free to use the logarithmic scale of decibans, where 10 decibans corresponds to 90% confidence, 20 to 99%, 30 to 99.9%, etc.)
This question freely acknowledges that there are aspects of physics which the atomic theory does not directly cover, such as conditions of extremely high energy. This question is primarily concerned with that portion of physics in which the atomic theory makes testable predictions.
This question also freely acknowledges that its current phrasing and presentation may not be the best possible to elicit answers from the LessWrong community, and will be happy to accept suggestions for improvement.
Edit: By 'atomic theory', this question refers to the century-plus-old theory. A reasonably accurate rewording is: "Do you believe 'H2O' is a meaningful description of water?".