There may be some disagreement as to where the precise climax of this haunting and mysterious poem lies, but to my mind the climax occurs in Part IV of the poem when the Mariner is finally able to rid himself of the sign of guilt that has hung around his neck since he shot the albatross and killed it towards the beginning of the narrative.
What occasions this reversal in his fortunes is his noticing of some water snakes. The Mariner clearly admires their beauty, their sense of life and their happiness, and as a result he feels love for them and blesses them:
O happy, living things! no tongue
Their beauty might declare:
A spring of love gushed from my heart,
And I blessed them unaware:
Sure my kind saint took pity on me,
And I blessed them unaware.
Notice how this action cancels out his act of destruction of Nature by killing the albatross. Having blessed living things, the Mariner suddenly finds that he is able to pray again. As a result, the albatross which symbolises his guilt and burden, falls away from his neck and into the sea, where it sinks, never to return:
The selfsame moment I could pray;
And from my neck so free
The Albatross fell off, and sank
Like lead into the sea.
This is the major resolution of the conflict that begins when the Mariner kills the albatross, which is why I think it marks the climax of the poem. The rest of the narrative charts the falling action and resolution of the story.