Actually, John 3 is saying that to be born again is to be born in the Spirit, which is what happens when we are saved.
3 Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. 4 Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his motherís womb, and be born? 5 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. 6 That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. 7 Marvel not that I said unto thee, Ye must be born again.
We don't have to die physically in order to be born spiritually. That's why there is such a continual war between the spirit and the flesh after accepting Jesus Christ as Saviour; the spirit is born again but the body (flesh) isn't. If we have the indwelling of the Holy Spirit, we have been born again and that happens before physical death.
I've always assumed that the bit about being born of water in verse 5 is referring to being baptised but I could be wrong about that.