The Gospel of the Grace of God centers on the idea of justification. What does it mean to be justified? It means "God declares you righteous". It's an acquittal.
Please don't make the mistake of thinking that means anything has changed in the one who is justified. When God declares someone righteous, He is simply saying "I won't hold your sins against you". That's it, that's all "justification" means.
When God justifies a sinner, that sinner is still a sinner. True, he's a sinner who will never stand condemned before God, but he's still a sinner. Justification doesn't make you a better person, it just means you've been acquitted. If you have been justified, the sins you committed---the big ones and the little ones---are still sins you committed. You don't un-commit those sins. But you have the assurance that God won't hold them against you.
And a sinner who's been justified doesn't suddenly stop sinning. But that doesn't affect her justification: God has declared that sinner to be righteous. The declaration has been made, and no sin before or after God has made the declaration affects what He has declared.
Now, the Scripture talks about something else, being born again. Let's be plain: I don't believe there is any Scriptural basis to say that justification and new birth happen separately. They happen at the same time, and I'm not about to weigh in on which one happens first. That's the sort of "angels dancing on the head of a pin" discussion that just gets in the way. But we need to understand that they are two distinct events. The believer has been justified, and he has been born again. But there are terrible problems for those believers who confuse the two. Perhaps we can discuss new birth another time, but for now, we'll restrict our discussion to justification.
Justification is a legal idea: it's an acquittal. It means God has dismissed all charges.
If I go out, having been justified, and I commit some heinous sin; that doesn't mean I'm no longer justified. God is righteous: He can't go back on His word and say "Well, I said I wouldn't hold your sins against you, but since you did that terrible thing, forget it! The deal's off!". God's not like that. God can't lie (Titus 1:2).
Assurance of salvation is based on our assurance of justification. This is an important point: there are too many Christians who look for assurance of salvation in their good works. They seem to think that they are justified by faith, but have to work to be sure. In contrast, Colossians tells us: "As ye have therefore received Christ Jesus the Lord, so walk ye in him" (Colossians 2:6, KJV, emphasis mine). As long as we're looking at ourselves, we'll never be confident that God won't condemn us: we are sure that we are justified because we are sure that God cannot lie.
I'm not trying to suggest that we ought not to have anything to show for having met God. I'm not trying to say we ought not to concern ourselves with whether we are living up to our calling. What I am trying to say is, we are sure we are justified because the Word of God declares that God justifies the ungodly.
Does this mean a true believer can sin? Absolutely! Does it mean we can just go ahead and keep on sinning, because we are guaranteed that we won't be punished for it? Yup. That's exactly what it means. Please understand this: anything less than a total and complete acquittal from all sins, past and future, wouldn't be enough. Anything less than that would leave a sinner doomed for an eternity in Hell. Because what the Word of God declares (and, less importantly, what every believer eventually experiences) is, we continue to fall short of the glory of God: "for all have sinned and fall short of the glory of God" (Romans 3:23, ESV). The sins and shortcomings will last until we finally meet Him face to face: any "gospel" that makes justification only apply to sins in the past is no gospel at all, but rather taunts the poor sinner for his inability to measure up.
Should a believer do that? Should someone who's justified just start a life of debauchery and wickedness, confident that God won't condemn him or her for it? Well, that's a different question. We'll get to Romans 6 later, but the answer is "no". You can: you can live a life of sin, and God won't go back on His word. But there are very good reasons not to: reasons that go far beyond the idea of punishment.
But as someone wiser than I am pointed out: if the question doesn't come up, you're not teaching the Biblical gospel.*
And in the end, if you have trouble with the idea that justification is based on what you believe, rather than on what you do; then you need to re-read Romans 1--4 again. "But to him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness."(Romans 4:5, KJV).
* Alan Gamble gave some messages at Greenwood Hills in 2005 entitled "Riches in Romans", in which he discussed this point. Unfortunately, they took that series from their website, no doubt to save space for the messages from 2006.