“the Law Of Sin”

What is your definition of "the Law?" If you mean the TC'S (Law of Moses), only the Hebrew/Jew had to die to it, which is what occurs at rebirth. Mankind is under the curse of "the law of sin," which was incurred to mankind from Adam (Rom 5:12). This law of sin has its force from God's command to Adam that, “thou shalt surely die” (Gen 2:17),” and when sin is finished,” which was at the time of his "transgression" (Rom 5:14; Gen 3:6), the curse of the law of sin and death came into effect.

God pronounced the law of sin and death on all mankind in Genesis 2:17. This became the law of sin, which states that "the soul that sins shall die" (Eze 18:4, 20). This curse was before the Mosaic Law (Rom 5:13) and was upon mankind. Therefore, in regeneration, the Hebrew/Jew is dead to the Law, and the law of sin, but the Gentile only to the law of sin. The curse of death continues in the body, but in the spirit, the guilt of sin is nullified (Col 2:14), and its dominion (Rom 6:12, 14) restrained, which restraint is commensurate with the believers "conformation."

Galatians 5:16: ". . . and ye shall not fulfil the lust of the flesh."
JG - “He does not say there shall be no flesh, nor any lust of the flesh in them if they walk spiritually; or that the flesh should not act and operate in them; or that they should do no sinful action; all which is only true of Christ; and the contrary is to be found and observed in all true Christians, though ever so spiritual.

"But that they should not fulfil or perfect the lust of the flesh; should not give up themselves entirely to the power and dictates of the flesh, so as to be under it and at its command, and be obedient servants and slaves unto it; for, in this sense only, such that are spiritual do not, commit sin, they do not make a trade of it, it is not their constant employ or course of conversation.”

The fulfilling of sin, same as bringing sin to completion, or same as "when sin is finished" (Jam 1:15) designs the fact of not being a prisoner or "captive" (Rom 7:23) to it, but rather a willing subject, who does not feel captive against his will. If your desire is to be where sin is, you’re not a captive (from captivated) because it is according to your will and desires.

All are indwelt by sin through the sin nature, but only through regeneration does one become an unwilling subject to it, which is the intention of the word "captive," denoting it’s against one's will. Hence Paul’s, “For what I am doing, I do not understand. For what I will to do, that I do not practice; but what I hate, that I do” (Rom 7:15).

The desire Paul maintains against the sin nature comes from regeneration, a condition which is void in an unbeliever. Romans Eight does not instruct the absence of the sin nature, but the freedom believers have from its condemnation (8:1, 33), and Romans Six from its dominion (6:12, 14). Unbelievers are not captives, but rather willing subjects, as we were before Christ. Thus, sin is in us, but we are not after it, e.g. "in it" (Rom 8:9).

Note: The term "flesh" in the NT in nearly all of its usages refers to the sinful nature ("old man," Adamic nature), not the physical body. Otherwise it would conflict with what Paul said concerning us not being in the flesh.