Adam was a natural earthy  man 1 Cor 15:45-49
45 And so it is written, The first man Adam was made a living soul; the last Adam was made a quickening spirit.
46 Howbeit that was not first which is spiritual, but that which is natural; and afterward that which is spiritual.
47 The first man is of the earth, earthy; the second man is the Lord from heaven.
48 As is the earthy, such are they also that are earthy: and as is the heavenly, such are they also that are heavenly.
49 And as we have borne the image of the earthy, we shall also bear the image of the heavenly.
I dont believe that, and i dont see any biblical reason why you would say something like that.
		
		
	 
The 
re-birth by the Holy Spirit is a 
second birth into a 
life Adam 
formerly had and lost in the fall; i.e., God's divine eternal life within Adam's immoral spirit, and 
now we are born without it, in 
spiritual death (which is 
not death of the immortal human spirit, but 
absence of divine eternal life within the immortal human spirit).
That divine eternal life of God is 
re-imparted to the human spirit in the 
re-birth (born 
again) by the Holy Spirit.
The human soul and human spirit do not occur separately in the NT, they are always together in the human person until death.
They share some of the same functions (emotion, rejoicing), and can only be distinguished there, not separated.
In reference to 
your Scripture above: in context of the resurrection and the resurrection body in 1 Co 15:35-58 and the contrast between the 
natural (sinful) and the 
spiritual (sinless) material body in the two Adam's, where
 
the first Adam had a 
natural body from the dust of the ground (Ge 2:7), and through whom a natural body is given to his descendants, and
the second (last) Adam, Christ, the life-giving spirit (Jn 5:26), who through his death and resurrection will at the second coming, give his redeemed people a 
spiritual body--
physical, yet imperishable, without corruption, and adaptable to live with God forever (Php 3:21),
it does not relate to our new birth, but to our two different bodies, pre- and post-resurrection.