A high luminosity muon collider requires single, intense, muon bunches with small emittances: just one of each sign. An efficient front end and a cooling channel have been designed and simulated within the collaboration of the Muon Accelerator Program. The muons are first bunched and phase rotated into 21 bunches, and then cooled in six dimensions. When they are cool enough, they are merged into single bunches: one of each sign. The bunch merging scheme has been outlined with preliminary simulations in previous studies. In this paper we present a comprehensive design with its end-to-end simulation. The 21 bunches are first merged in longitudinal phase space into seven bunches. These are directed into seven ``trombone'' paths with different lengths, to bring them to the same time, and then merged transversely in a collecting ``funnel'' into the required single larger bunches. Detailed numerical simulations show that the 6D emittance of the resulting bunch reaches the parameters needed for high acceptance into the downstream cooling channel.
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