Actions have consequences, that's just common sense. Western religions believe in rebirth in hell or heaven. Islam believes you are reborn from hell to heaven when the punishment is complete. The best explanation comes from Western metaphysics, every moment the body changes. It is discontinuous. In Buddhist terms, every moment is a rebirth of the previous moment. Western medicine has M.D. accounts of patients reporting events when they were clinically dead.
Bottom line. There is no Buddhism without both karma and rebirth. Without karma and rebirth, Buddhism would be completely useless and meaningless. In sum, Buddhism is the teaching about karma and rebirth.
You are calling impermanence (the momentary existence) of all things rebirth. What impermanence really means is that there is no abiding self in things. There is no self in things, so how could that self be reborn from instant to instant?
The concept of rebirth doesn't go against the idea of anatman. The concept of reincarnation, however, does, because it assumes that there is a permanent entity like a soul traveling through time and taking different bodies. This is not the same as rebirth. You don't have a self going from this moment to the very next that has more than conceptual existence. However, to the unenlightened so it seems. Although there's no real self going through this life, we act as if it did. We live as if it did, die as if it did and take a new birth as if it did. What there is is a mental continuum, thoughts and feelings perceived by an awareness unsubstantial that doesn't stop when this body dies. This deep awareness can be found through meditation and there are experiments which can be performed regarding the investigation of its starting point by the practitioner.
Rebirth is just a change in circumstances a little deeper than usual. But in essence, life, death, rebirth and so on are deluded perceptions.
Ron hints that in his original way of saying things. Truly there's no life nor death and these are but deluded experiences. To gain an insight about this is, IMO, quite important, but also not that easy for everyone. Gaining a bit of insight about this, also lightens a little our concerns regarding rebirth and karma. These teachings are quite useful, but not the main point. They are useful especially when we check our conduct, but even then, the tendency will be going from sets of rules to a behavior guided by wisdom. When one investigates the mind deeply enough, one will also realize that even a dramatic set of circumstances such as death doesn't put an end to consciousness. It's just a change, a little faster than the usual changes which are constantly occurring all around. i.e., I had a body in the past that is no more, only a result remains, my adult body. I had feelings and ideas in the past that are no more, only this awareness of what comes and goes. I'm only the result of a person who doesn't exist any longer. If my memory fails me and I suffer from amnesia, I'll still be the result of that person, even though I don't share a single atom or a single mental process with him. Just this awareness that watches changes going on, experiences which are now bright and vivid and will fade away in the future to the point of oblivion.
Sometimes what we need to try to understand is what is blocking us from a deeper understanding about rebirth. What makes us prone to dismiss this idea without first giving it proper thought and putting it to the test? Sometimes we may discover that even without noticing we have a predilection for a certain system of metaphysical assumptions we've assimilated during our education. We are tired of knowing that what seems is not always what is and even though it seems the sun goes around the sky, Earth revolves around the sun, even though it seems we die and that's it, there may be the case that things are different. Episodes of people claiming to recall past lives are not rare nor recent. Are they imagining things? I don't think such is always the case.
Our culture encourages parents to repress recounts of past lives memories if their offspring manifests them. So it is natural that we don't hear about it often, unless we dig a little. Were you living in Tibet or in India and probably you'd face a whole different set of beliefs.
We tend to be a little ethnocentric in our worldview, and that is natural. These days such problem is being lessened, one of the positive effects of globalization (it had to have something good, for it has so many bad). There were times when according to us proper people were white and Christian. Otherwise they were considered barbaric, sometimes not even human.
Well, I'm diverging. Anyway, the point is, it is worth a while to try methods that some people say will allow you to check for yourself rebirth, karma and so on. Choosing what is not more than a temporary set of metaphysical predilections in the guise of scientific facts doesn't seem productive. Science may have a word about many, many subjects, but for the time being it has no answers to these questions, at least answers properly built according to good methodological practices and intellectual honesty. So, in the future perhaps, but for now it's a dead end. Then you have Buddhadharma and its propositions. One of them, taken quite seriously, is that being a good honest person is part of the way to go. If you meet closely some real practitioners, you'll be impressed with their kindness, honesty and so on. That doesn't go along well with lying about knowing the truth about rebirth. So perhaps if we listen to the experts in this field, first understanding why we can consider them so, and when we put traditional scholastic divergences apart, we will see that they all pretty much speak the same language. Then we test it ourselves as, so far, we don't seem to be able to get answers from anywhere else but our own experience. Seems an interesting proposal to me.