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A bargaining framework and a measure of conflict polarization are developed from two elements: (i) hatred-based negative externalities experienced by the parties to the conflict, and (ii) penalties the parties impose on their delegated negotiators when concessions are made in the bargaining process. The framework establishes agreement and disagreement regions and it is shown that a necessary condition for a negotiated solution is the adoption of a dual policy that combines dissociative political and military strategies. This analytical approach is applied first to polarized conflicts generally and then to the specific case of the internal conflict in Colombia between that country's government and the FARC guerrilla group. The model provides a rationale for the complex dynamic of Colombia's current peace process, which has involved a preliminary agreement and its subsequent rejection in a national referendum. Our analysis highlights the successful dissociative political-military strategy followed by the negotiators that enabled them to reach the agreement and the negotiators' underestimation of the hatred levels that led the majority of the Colombian society represented in the referendum to vote the agreement down because they considered the concessions made by the government too generous to be acceptable.