This Sunday's second reading from Hebrews professes:
Just as it is appointed that human beings die once,
and after this the judgment, so also Christ,
offered once to take away the sins of many,
will appear a second time, not to take away sin
but to bring salvation to those who eagerly await him.--Hebrews 9
And this is what the Catechism of the Catholic Church has to say about reincarnation:
Death is the end of man's earthly pilgrimage, of the time of grace and mercy which God offers him so as to work out his earthly life in keeping with the divine plan, and to decide his ultimate destiny. When "the single course of our earthly life" is completed, we shall not return to other earthly lives: "It is appointed for men to die once." There is no "reincarnation" after death.--CCC 1013
CCC Search Result - Paragraph # 1013
1013 Death is the end of man's earthly pilgrimage, of the time of grace and mercy which God offers him so as to work out his earthly life in keeping with the divine plan, and to decide his ultimate destiny.
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There are not a few Catholics, especially in the world of Internet discussions, who assert, "Well, I'm Catholic but I believe in reincarnation".
Unfortunately, Catholics are not free to choose what teachings on faith and morals they accept or reject. They can't assert this any more than someone can say, "Well, I'm Catholic but I believe that the Epistles of Paul don't belong in the Bible" or "I believe that Mary is a goddess, and I'm Catholic" or "I believe that women should be ordained, and I'm a lifelong Catholic".
We have the Faith, given to us once for all (Jude 1:3), and we must conform ourselves to it.
Not create a faith that conforms to our own person views.
If we are in a discussion regarding reincarnation, a good question to ask is, "What evidence do you have for reincarnation?" Whenever someone makes a positive assertion (i.e. "Reincarnation is true!"), then the the burden of proof for an unsubstantiated supposition falls on the person making the assumption.
Some proponents of reincarnation propose that the Bible supports it, specifically, they will cite Matthew's gospel where Jesus indicates that John the Baptist is the reincarnation of Elijah the Prophet:
And the disciples asked him, "Then why do the scribes say that first Elijah must come?" He replied, "Elijah does come, and he is to restore all things; but I tell you that Elijah has already come, and they did not know him, but did to him whatever they pleased. So also the Son of man will suffer at their hands." Then the disciples understood that he was speaking to them of John the Baptist.--Matthew 17:10-13
Apologist Tim Staples responds to this:
"Jesus is not speaking of reincarnation when he speaks of “Elijah [having] already come.” He speaks of St. John the Baptist having the spirit and the power of Elijah. In fact, Luke 1:16-17 helps us to understand Matthew 17:10-13 better when the angel Gabriel gives us further definition, if you will, of what "Elijah [having] already come" actually means. He says to Zechariah, the father of St. John the Baptist, concerning his son who would soon be miraculously conceived:
And he will turn many of the sons of Israel to the Lord their God, and he will go before him in the spirit and power of Eli'jah, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just, to make ready for the Lord a people prepared.
Notice, he comes “in the spirit and power of Elijah” as a prophet of God. That is what is meant by “Elijah has already come.”http://www.catholic.com/blog/tim-staples/is-reincarnation-biblical.
Also, logically, the idea of reincarnation doesn't make sense. What mistakes in our past lives have caused us to suffer in this life? If we are to learn from our past mistakes, then why can't we remember what our past mistakes were? What's the point of being reincarnated if we can't remember what we did wrong in the previous life?
Finally, reincarnation is not compatible with Catholicism because reincarnation views the body as a mere receptacle--a shell to hold the spirit. In reincarnation the spirit is supreme; the body is insignificant.
However, in Catholicism, the body is wondrous, awesome and profoundly important! We are not merely shells holding an immortal soul, but a magnificent Body-Soul composite
From EWTN: For a Christian, the body's significance is good, inescapable, and central; Christianity itself cannot be understood apart from an appreciation of the body. It is a myth that the Catholic Church teaches as it does about sexuality because it undervalues sex. The Church teaches as it does because it values human sexuality so highly. And in valuing sexuality, it necessarily values the body.
A PASTORAL LETTER