1988 Ford Mustang Melted Negative Battery Cable.

Just driving down the stret and all of a suden my 1988 2.3ltr Mustang stops. Raise the hood and the neg battery cable is melted. I have continuity from the + to the - terminals. I replaced both + and - cable and still no start with a hot + cable now. Replaced the starter and the car started but would not shut off. Replaced the solinoid and car started, ran and shut off fine. Went out later and the ignition is completely dead with a fully charged battery. I have continuity from the + to the - terminals again. To sum up. Replaced all wired from the battery, starter, solinoid and have a fully charged battery. Help please.