A bad fuse won't case the battery to show no voltage across the terminals, I think it is most likely a bad battery. Even a battery showing 12.7V should be enough to crank it at least some not go to zero volts when you try and start it. And for it to happen so suddenly I would imagine you have a shorted cell. That is as long as I understood you correctly, you measured voltage with the probes on the battery terminals themselves,correct?

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