No. The death of Achilles is not told in the Iliad, but it is foretold several times, and he knows that it is near. In Book I, his mother, Thetis, tells him: "your portion of life is brief, not at all long." In Book XVIII, after the death of Patroclus, she repeats that soon after Hector his own end will be ready, and Achilles answers: "let me die now," and he decides to avenge his friend all the same.
In Book XIX, the horse Xanthus, to whom Hera gave a voice, warns that "the day of ruin is near you" and that the blame will not be his, but that of "a great god and the strong Moira." Achilles answers: "I myself know it well: my fate is to perish here, far from my dear father and mother; but even so, I will not stop until I have made the Trojans glut themselves on war." In Book XXI, as he kills the Trojan Lycaon, he speaks of his own death: "there will be a dawn, or an evening, or a midday." In the same book, caught in the current of the river, he says that his mother deceived him when she promised that he would die beneath the wall, by the arrows of Apollo, and that he would rather have been killed by Hector.
The most precise prophecy comes from Hector, who dies in Book XXII warning that Paris and Apollo will destroy Achilles beside the Scaean Gates. Achilles answers: "die; as for my Ker, I will receive it when Zeus and the other immortal gods wish to bring it about."
The poem keeps death outside the scene. Achilles stands before it all the time and, even so, chooses glory over return, as Book IX had already shown. Readers argue over whether the choice is free or whether he only confirms what fate has already fixed. The text says both.