Yes - it's the first moment that the audience can be sure that Claudius is in fact guilty, and that the ghost (even if not the spirit of Hamlet's father) is honest. Until then, nothing has been certain.
O, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't,
A brother's murder! Pray can I not,
Though inclination be as sharp as will;
My stronger guilt defeats my strong intent,
And, like a man to double business bound,
I stand in pause where I shall first begin,
And both neglect. What if this cursed hand
Were thicker than itself with brother's blood,
Is there not rain enough in the sweet heavens
To wash it white as snow?
No ambiguity there! Hope it helps!
Claudius is overcome by guilt in Act lll, Scene 3. Before he kneels down to pray for redemption, he utters a long-winded monologue in which he confesses his crime. Claudius states that his crime is so terrible that its stench rises to heaven and that it is equal to the first capital crime -- Cain murdering his brother Abel. He states that he cannot pray, for his guilt is too great. He rhetorically asks whether he can cleanse his hands of his terrible deed with rain from heaven and make them pure again. He concludes that, to receive mercy, he first has to confront his misdeed and acknowledge guilt.
Claudius states that he can look up to heaven and his fault will be forgiven, but he is unsure which form his prayer should take. He asks whether he can ask for forgiveness if he is still enjoying the fruits of his crime. He admits that, at times, the beneficiary of such a criminal act rises above the law and can avoid retribution (as he has, so far) but in heaven, one cannot. It is obvious that in heaven Claudius believes people present their true selves and are not protected by their earthly authority.
This presents Claudius with a conundrum. When one is compelled to admit wrongdoing and give evidence, what kind of repentance would there be? What would happen if he cannot truly repent? Claudius feels wretched about his situation, and emphatically cries out that his bosom is "as black as death" -- his heart is possessed by evil and he is battling to free his conscience. He is distraught that he cannot free himself from the guilt that torments him. The more he wants to be released, the more bound he becomes.
Claudius then makes an impassioned cry to the angels to help him and determine the true nature of his character. He literally forces himself to kneel and begs his hard heartstrings to soften so that he may metaphorically achieve the innocence and kindness of a newborn. Once he has achieved this, there may be a chance for absolution. He then proceeds to kneel down and pray.
In Act lll, Scene 1, Claudius alluded to his guilt after a remark made by Polonius, his chief advisor and charge d'affaires:
Polonius stated, in part, "that with devotion's visage/ And pious action we do sugar o'er/ The devil himself," to which Claudius responded in an aside:
O, 'tis too true!
How smart a lash that speech doth give my conscience!
The harlot's cheek, beautied with plastering art,
Is not more ugly to the thing that helps it
Than is my deed to my most painted word:
O heavy burthen!
Claudius admits his speeches and compliments are, just as a prostitute's make-up, not enough to hide the ugliness beneath which, in this instance, is, of course, his crime.
Claudius is in a small prayer chamber where he has gone after watching "The Murder of Gonzago". It was common for king's to have their own private prayer rooms so that they could pray in private. Hamlet, knowing the location of the king's prayer room, also knows a way in which to watch him pray. In many stage adaptations, the chamber also has a confessional box for the priest and that is where Hamlet goes to overhear Claudius' confession.