Hebrews 9:27, as cited by Caleb in his answer, is the single Bible verse that succinctly rejects Reincarnation for a person holding a Christian worldview. However, to really tackle the issue, you need to know what people mean by their terms.
In that famous Hindu text, THE BHAGAVAD-GITA, Chapter 2, we have the Song Celestial. Here are a few words:
As the wise bear. The soul which is not moved,
The soul that with a strong and constant calm
Takes sorrow and takes joy indifferently,
Lives in the life undying! That which is
Can never cease to be; that which is not
Will not exist. To see this truth of both
Is theirs who part essence from accident,
Substance from shadow. Indestructible,
Learn thou! the Life is, spreading life through all;
It cannot anywhere, by any means,
Be anywise diminished, stayed, or changed.
But for these fleeting frames which it
informs With spirit deathless, endless, infinite, They perish.
Let them perish, Prince! and fight!
He who shall say, “Lo! I have slain a man!”
He who shall think, “Lo! I am slain!”
those both Know naught!
Life cannot slay. Life is not slain!
Never the spirit was born;
the spirit shall cease to be never;
Never was time it was not; End and Beginning are dreams!
Birthless and deathless and changeless remaineth the spirit for ever;
Death hath not touched it at all, dead though the house of it seems!
Who knoweth it exhaustless, self-sustained, Immortal, indestructible,--
shall such Say, “I have killed a man, or caused to kill?”
The affirmation "Life is not slain" is a focal point of disagreement between Christianity on the one hand and Buddhism, Hinduism, and other religions that believe in reincarnation. In such light, the Hebrews verse, which declares a final judgment, only challenges Hinduism with its endless cycle, not Buddhism, which reaches forward to an end to the cycle of rebirth, to Nirvana. One who believes that is quite comfortable with both Hebrews 9 and Reincarnation. There is only one death, one end to this existence, and some process of judgment determines when we are prepared spiritually for that transformation.
For a Christian to say that he or she rejects reincarnation because of the Bible's teachings, you need to nail down the meanings of words like "birth", "death", "judgment", "see", "eternity" and "time". One book in the Bible tackles that head on: Ecclesiastes.
Time: Cyclical or linear?
Reincarnation dictates a cyclical view of time. Ecclesiastes begins in chapter 1 by positing a cyclical view of time as well, with no essential progress being made by mankind.
3 What do people gain from all their labors
at which they toil under the sun? 4 Generations come and generations go,
but the earth remains forever. 5 The sun rises and the sun sets,
and hurries back to where it rises. 6 The wind blows to the south
and turns to the north; round and round it goes,
ever returning on its course. 7 All streams flow into the sea,
yet the sea is never full. To the place the streams come from,
there they return again.
This cyclical view is repeated in chapter 3:
3 There is a time for everything,
and a season for every activity under the heavens:
2 a time to be born and a time to die,
a time to plant and a time to uproot,
3 a time to kill and a time to heal,
a time to tear down and a time to build,
4 a time to weep and a time to laugh,
a time to mourn and a time to dance,
5 a time to scatter stones and a time to gather them,
a time to embrace and a time to refrain from embracing,
6 a time to search and a time to give up,
a time to keep and a time to throw away,
7 a time to tear and a time to mend,
a time to be silent and a time to speak,
8 a time to love and a time to hate,
a time for war and a time for peace.
But then the Teacher (or Preacher) says something new:
9 What do workers gain from their toil? 10 I have seen the burden God
has laid on the human race. 11 He has made everything beautiful in its
time. He has also set eternity in the human heart; yet no one can
fathom what God has done from beginning to end.
The Teacher declares that there is a beginning and an end. See what is different between the Bible and the world's other religious writings? It reveals an undeniable cyclical pattern that we observe with our eyes, then tells us that we cannot see the essential reality. Eternity cannot be seen with our eyes. God put it in our heart but we cannot understand it.
Seeing or Believing?
Ecclesiastes contrasts life lived "under the sun" with that "under the heavens". Seeing is observing with eyes, ears, nose, mouth, and hands. Our purpose in life and the eternal reality that we seek cannot be seen. It is beyond our observational capabilities. Among the purposes that are not eternally significant, that are futile, are: work, romance, possessions, religion, intellectual pursuits, fame, power, pleasure, and madness.
What is given as the cause of the problem? Death. Death interrupts our activities. Death blocks our sight.
12 For who knows what is good for a person in life, during the few and
meaningless days they pass through like a shadow? Who can tell them
what will happen under the sun after they are gone? (Ecclesiastes
No one knows what is coming—
who can tell someone else what will happen after them? (Ecclesiastes 10:14)
Death is given as an impenetrable barrier. How can one "see" their past lives as some who believe in reincarnation claim? This path to knowledge is ruled out by the Teacher.
Remember your Creator
in the days of your youth... (12:1)
We are created beings, hence we have a beginning. There is "a time to be born and a time to die" (Ecclesiastes 3:2). Not multiple times - one time for each of us.
Finally, Ecclesiastes ends with a final announcement of judgment. The cycles may govern our mechanical world, but our souls have a definite beginning and an end in judgment.
You who are young, be happy while you are young,
and let your heart give you joy in the days of your youth.
Follow the ways of your heart
and whatever your eyes see,
but know that for all these things
God will bring you into judgment. (Ecclesiastes 11:9)
13 Now all has been heard;
here is the conclusion of the matter:
Fear God and keep his commandments,
for this is the duty of all mankind.
14 For God will bring every deed into judgment,
including every hidden thing,
whether it is good or evil. (Ecclesiastes 21:13-14)
Ecclesiastes is mute about resurrection, but Job is not.
25 I know that my redeemer lives,
and that in the end he will stand on the earth.
26 And after my skin has been destroyed,
yet in my flesh I will see God;
27 I myself will see him
with my own eyes—I, and not another.
How my heart yearns within me! (Job 19:25-27)
Thus in Ecclesiastes you find an entire worldview elaborated in which time appears cyclical but is not; it is linear and we get but one life.