Yes, he does. Every word. It may seem to casual viewers, though, that some lines are omitted, because Branagh transposes them to a different moment in the play. For instance, Hamlet's first words to the Ghost - "Angels and ministers of grace defend us! Be thou a spirit of health, or goblin damned, bring with thee airs from Heaven or blasts from Hell, be thy intents wicked or charitable, thou comest in such a questionable shape that I will speak to thee", etc. , meant to be spoken just before the Ghost beckons him to follow, are switched to a few moments after that, during which he is shown running through the forest after the Ghost. As he runs, we hear Hamlet's words in voiceover, then he stops and says "Whither wilt thou lead me? Speak. I'll go no further", and the Ghost speaks for the first time.
And at the very beginning of the film's second half, we hear Claudius (Derek Jacobi) in voiceover saying "O Gertrude, Gertrude, when sorrows come they come not single spies but in batallions" -lines that he normally says after we have seen Ophelia insane, not before. Claudius's prayer scene is also transposed to just after Hamlet's soliloquy " 'Tis now the very witching time of night", when in the play it occurs several lines before that.