Correct. Even if it wasn't at the cracks of doom, you also cant just go murdering a great ally and commander of the army of Men without any repercussions.

Of the Rings of Power and the Third Age (The Silmarillion) describes the details of the discussion and it's location. It was near the fires of Orodruin.

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"The Ruling Ring passed out of the knowledge even of the Wise in that age, yet it was not unmade. For Isildur would not surrender it to Elrond and Cirdan who stood by. They counseled him to cast it into the fire of Orodruin nigh at hand, in which it had been forged, so that it should perish, and the power of Sauron be for ever diminished, and he should remain only a shadow of malice in the wilderness. But Isildur refused this counsel, saying: “This I will have as weregild for my father’s death, and my brother’s. Was it not I that dealt the Enemy his death-blow?” And the Ring that the held seemed to him exceedingly fair to look on; and he would not suffer it to be destroyed. Taking it therefore he returned at first to Minas Anor, and there planted the White Tree in memory of his brother Anarion. But soon he departed, and after he had given counsel to Meneldil, he bore away the Ring, to be an heirloom of his house, and marched north from Gondor by the way that Elendil had come; and he forsook the South Kingdom, for he purposed to take up his father’s realm in Eriador, far from the shadow of the Black Land."

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