Well we could have had an amplifier there that was bypassed (they fail a LOT). The box will be mounted on the floor, not to the seat, in case that was unclear. In either case the solution would still be the same.
For the sake of safety and the easiest solution, you are going to want to purchase an adapter kit for this radio which will avoid needing to cut any wiring. The adapter will connect your primary plug from the radio to the new radio's plug. The smaller secondary plug from the old radio will not be used (it is for the display on the top of the dash which only works with the factory radio).
You will likely have two antenna jacks on your old radio (the metal bullet cables). The larger one you will plug into the new radio, the smaller one you will simply tape up. Certain models had dual antennas, one fender antenna and one glass antenna, others had one or the other. If you have both, use the large one that will fit directly and just tape up the smaller. If you only have one antenna, it will be the large one anyway and will just plug in to the new radio.
The round DIN cable is the feed for the amplifier. You will plug this into the adapter that converts the DIN cable to four RCA jacks, which will plug in to your new radio.
Here is an example of the cable you need:http://www.crutchfield.com/p_120707003/Receiver-Wire-Harness.html?tp=2977
Note that it says for "Infinity System" as you indicated originally. This is so that it includes the DIN adapter for the amplifier wiring.
If you did NOT have the Infinity System, the kit will be a different one that does not come with a DIN adapter and the main plug has more wires (used for speaker wires not used on amplified systems).
Get the adapter kit and life will be much easier... just match up the wire functions from the adapter kit to the same functions on the new radio (both will have instructions showing each) and there will be nothing to cut etc.