The contradiction occurs when we assume there is soul which is reincarnated, as this contradicts the actual Buddhist teachings on 'anata' which roughly translates as no-self.
In Buddhism what we think of as an individual is called the 'santana', which could think of as a mind/body for any practical purpose. The santana is thought of as a transformation which is eternal in its own right - but it has no 'identity', so is still anata.
'Rebirth' is a complicated concept, because it refers to the santana in a sense, but mainly only when identity is attached to it, which is grasping, clinging, craving etc. These latter qualities are called 'the volition', and this is what perpetuates rebirth.
'Volition' can be thought of as any urge to move the mind, and it is also categorised into good-will and ill-will, but I don't think that's particularly relevant to this topic, so maybe it'll come up later on. It's just because volition is the cause, or the creation of potentials which manifest in lived experience at such time when the conditions for it to arise are right. Hence they say good-will produces potentials for good outcomes, and the opposite is true of ill-will.
What they call 'liberation' is the cessation of volition, but for many Buddhists this isn't considered full enlightenment because some potentials created by past volitions have not yet arisen, as they must, as lived experience. No new potentials are created but the old potentials generated by past volitions still continue to arise to conscious experience and dissolve away. The expiration of the potentials without generating new ones is how the 'purification' functions.
When all the old potentials have all expired, then there is 'full enlightenment'. Some maintain that this can only happen when the physical body dies.
So to sum up, the volition doesn't operate, and the potentials have all expired, so there no rebirth. Thus the cycle of birth and death ceases.
In the deeper philosophy, there never was anyone born or dying. That was only a delusion perpetuated in ignorance - and this would lead us to the teachings on 'dependent arisings' (the first of which is 'ignorance').
Its a very subtle meaning and I doubt anyone really understands it, but I think it is best thought of as any urge to move the mind. We could say it's completely delusional in essense, because the mind is always still until it goes into reactivity. Then we do a very deep analysis of the body sensation and the mind's reactions in aversion to discomfort and desires for pleasure. This is to say, if one ceases the psychological reactivity, the mind is still and quiet. So, in this sense, the cessation of reactivity is also the cessation of volition.
I think there is a good explanation here.
http://www.buddhanet.net/nutshell09.htm