The Bible teaches that we are born once physically and die once physically, and after that comes the judgment:

Hebrews 9:27
"Just as man is destined to die once, and after that to face judgment."

In the spiritual sense, the Bible says that we must be born twice but not reincarnated:

John 3:3
In reply Jesus declared, "I tell you the truth, no one can see the kingdom of God unless he is born again."

1 Peter 1:23
For you have been born again, not of perishable seed, but of imperishable, through the living and enduring word of God.

Titus 3:5
he saved us, not because of righteous things we had done, but because of his mercy. He saved us through the washing of rebirth and renewal by the Holy Spirit ...

Jesus taught that a person who is born again in Christ dies once (physical death) but never faces spiritual death:

John 11:25-26
Jesus said to her, "I am the resurrection and the life. He who believes in me will live, even though he dies; and whoever lives and believes in me will never die. Do you believe this?"

John 5:24
(Jesus is speaking) "I tell you the truth, whoever hears my word and believes him who sent me has eternal life and will not be condemned; he has crossed over from death to life.

Meanwhile, the Bible says a person who is born once spiritually and is never born again dies twice (both physically and spiritually):

Revelation 21:8
"But the cowardly, the unbelieving, the vile, the murderers, the sexually immoral, those who practice magic arts, the idolaters and all liars--their place will be in the fiery lake of burning sulfur. This is the second death."

That's why you sometimes see bumper stickers that say:

Born Twice, Die Once
Born Once, Die Twice

H/T Apologetix FAQ