The defilement of Adam's nature was physical. We all inherit this physical defilement, Scripturally called "sinful flesh" or "flesh full of sin" (Rom. 8:3). This physical law of sin in our bodies is not moral guilt or transgression. We are guilty of our own sins, not Adam's sin. Still, we all inherit this physical sin from Adam. The inheritance of this physical nature from Adam is our misfortune, not our crime.
The inheritance of this law, or physical sin in our bodies, made it possible that God, (in His role as "God our Saviour" Titus 3:4), could condemn "sin in the flesh" in the body of Jesus who was born with our identical nature. He could exhibit sin as terrible and worthy only of death as to the body of Jesus, while at the same time bring about life through the moral perfection exhibited in the life of Jesus.
What was it that the law (of Moses) could not do? The law could not give life "through the weakness of the flesh." The law could have given life, if man could have kept the law. The Apostle Paul called the Mosaic Law "the commandment, which was ordained to life" (Rom. 7:10). Jesus also, when asked "what must I do to inherit eternal life" told the lawyer, "What is written in the law? how readest thou?" (Luke 10:25). But through the weakness of the flesh, man could not keep it. Instead of being a commandment to life, the real life experience of men was that it was a commandment unto death.
Since men could not obtain life through the Mosaic Law, God provided a way. God sent Jesus in the likeness of sinful flesh. The word "likeness" really means indentical-ness. God sent Jesus in the same nature as all mankind bore--sinful flesh. He then condemned sin in the flesh of Jesus.
John Thomas, founder of the Christadelphian movement in the 1800s, explained this verse in the most easy to understand terms. "Sin could not have been condemned in the body of Jesus, had it not existed there."
Jesus lived a perfect life, and morally he was sinless: but, the nature that he inherited from his mother Mary was the identical nature to all mankind which was sinful--full of sin. When Jesus went up on the cross, the sinful body was crucified and destroyed; upholding God's condemnation of sin. But his perfect sinless life made it impossible that the grave could keep him. After three days, he was resurrected to life, becoming the first fruits of those who sleep, and opening the way to life for us by forgiveness of our sins through his great sacrifice.