Is there proof that rebirth does not exist?That it is not real?
There was a North American academic named Ian Stevenson who spent decades investigating people who claimed to remember their previous lives.
Some people cite him as having found evidence of rebirth.
In summary, he sought and collected stories, and discarded the stories which were proven untrue or fabricated. After discarding the untrue stories, what he had left were the stories which he didn't prove were untrue (i.e. stories which appeared to be true).
The Skeptic's dictionary entry on that subject says there's no way to use that methodology to prove that rebirth doesn't happen,
The historian, the journalist, and the jury member, on the other hand, must constantly deal with issues like lying, hoaxing, and fraud. Most important, though, is the fact that historians, journalists, and jury members have to evaluate the words and perceptions of people, rather than the structure or properties of things. Stevenson's work would require constant vigilance against being deceived by his subjects. Furthermore, since we know that people can have memories and be completely unaware of the source of those memories, he would have to be vigilant in identifying which memories were likely the result of cryptomnesia. Also, there is the major problem of providing an explanation for how a personality can survive death and transfer to another body, something Stevenson had no answer for. Finally, the most problematic issue Stevenson would have to face using his method of collecting stories would be the fact that nothing could ever count against his hypothesis. Stories that are rejected as hoaxes, frauds, questionable, unreliable, or based on experiences in this lifetime would be discarded, but they wouldn't count against the reincarnation hypothesis. The worst case scenario for Stevenson's method would be that his evidence does not compel belief and that even the best of it is open to alternative interpretations. Unfortunately, that is also his best case scenario. Most people are not likely to be too impressed when they realize that all Stevenson had to show for over forty years of research is that it is now false to claim that there is no evidence for reincarnation. It is still quite reasonable, however, to claim that there is no compelling evidence for reincarnation.
Stevenson would conduct dozens of interviews and spend hours searching through hospital and court records, trying to establish that there was no fraud involved, that the story wasn't contaminated, that there weren't errors in translation, that the events weren't just coincidental, that the child couldn't have gotten the information in any normal way. When he satisfied himself that there was no normal explanation for the concordance of story and facts, he would count the case as "solved" and see it as a piece of positive evidence supporting the reincarnation hypothesis. If he got a PLE story but couldn't corroborate it with facts, he called the case "unsolved." There is nothing that could be discovered by this method that could ever falsify the reincarnation hypothesis. And it remains a mystery as to what further research, that might be falsifiable, could ever evolve from Stevenson's technique.
Based on this Skeptic's dictionary entry, IMO there is no "proof of non-rebirth" (otherwise they might have referenced that proof in the Skeptic's dictionary). The most they can say, their strongest argument against his work, is, "There is nothing that could be discovered by this method that could ever falsify the reincarnation hypothesis."
In other words the skeptic seems to me to be claiming that the theory is, not "proven untrue", but rather "unfalsifiable" ... perhaps analogous to the way in which 'existence of God' or 'Russel's teapot' are unfalsifiable.
Though rebirth cannot be proved scientifically (because we don't have the tools at the present state of technology), there is a lot of anecdotal evidence and it can be proved with experience.
As I've mentioned in an answer to another question, there were many people in known history, and some even living in the present times who have recalled up to the minutest detail, their past life and its environment.
Forget even that, the most important proof for an individual is experience itself. If you perform vipassana regularly and attain the higher jhanas, you will be able to recall your past-life with your own insight (pradnya). There will be no need of any proof then!
Scientifically speaking no. As you mentioned "proof" I assume you want a sort of scientific answer.
Science works with what can be proved either by experience or by mathematics, there is no way rebirth can be proved through calculation or a controlled experiment, at least in current science.
There are cases of people that claim to remember past lifes, however, scientifically speaking they are not accepted as proof.
Technically speaking science cannot prove that it exists or doesn't exist, but when in doubt scientists always go for the simplest answer.
In Buddhism we accept past lifes and rebirth because they are a key part of the religion, the Buddha said right after he attained enlightment:
"I, who have been seeking the builder of this house (body), failing to attain Enlightenment which would enable me to find him, have wandered through innumerable births in samsara. To be born again and again is, indeed, dukkha! Oh house-builder! You are seen, you shall build no house (for me) again. All your rafters are broken, your roof-tree is destroyed. My mind has reached the unconditioned (i.e., Nibbana); the end of craving (Arahatta Phala) has been attained."
Yes. But that testimony need eyes to see! Not with your eye, but with your heart.
Everyone has to unfold his own mystery! One can point out what cold means, but he himself has to experience it to know it. Mere naming cold wont do! Proof of rebirth and no-rebirth both exist in eye of wisdom!