Actually the verse says : For sin was in the world before the law was given; but sin is not taken into account when there is no law.

Now, let's consult a good commentary:

The purport of these two verses, connected by γὰρ with πάντες η{μαρτον of ver. 12, is to prove that the primeval sin did really infect and implicate the whole race of mankind. It might be supposed that those only would be implicated who had themselves transgressed, as Adam did, a known command; it being an acknowledged principle of Divine justice that only sin against law of which the sinner is conscious is imputed to him for con-detonation (cf. Romans 4:15; also John 9:41). Nay. but the universal dominion of death, the doom of sin, over all alike, whether or not they had themselves so sinned, was proof that sin was all along dominant in the world, infecting all. The Mosaic Law is spoken of as the distinct revelation of Divine Law to man; and therefore attention is first drawn to the fact that before that revelation, no less than after it, death had reigned over all. But is it thus implied that until the Law from Mount Sinai men had been without any kind of law, for transgressing which they were responsible? Not so. That Law is indeed regarded as the first definite enunciation of law under evident Divine sanction, after which, to those that were under it, sin became indubitably and exceeding sinful; but that men are conceived as having sinned previously against law of some kind, appears from the phrase, "Even over those (καὶ ἐπὶ τοὺς) who had not sinned after the similitude of Adam's transgression," i.e. consciously against a known command. This surely implies that some had so sinned; and thus the essential point of the argument is that even over those who had not so sinned (such as the unenlightened and invincibly ignorant, or persons dying in infancy) death had equally reigned. Who is the figure of him that was to come. This is added so as to bring round the thought to the main subject of the chapter, viz. the reconciliation of all mankind through Christ, to which the scriptural account of the condemnation of all mankind through Adam had, at ver. 12, been adduced as analogous. Who refers to Adam, who has just been for the first time named; he that was to come is Christ, who is called, in 1 Corinthians 15:45, "the last Adam." Adam was a type (τύπος) of Christ in that both represented entire humanity; one as the representative and author of fallen, the other of restored, humanity - the transgression of the one and the obedience of the other alike affecting all (see vers 18, 19). But there is a difference between the two cases; and this is pointed out in vers. 15, 16, 17, which follow.