First let me state that I am NOT an electrician.
BUT my brother is and he taught me enough for my personal use and since I have wired over 3000 feet of wire in my house, a sub panel, receptacles, switches, lights, etc and I have NEVER encountered anything with only 1 wire.
Electricity flows in a circuit which by definition needs a wire to enter and a wire to leave.
Some old school electricians did something called "breaking the neutral" where they carried load on the white (neutral) wire but that is no longer acceptable as it charges your neutrals & at the panel also.
So, like a previous poster said, there has to be another wire somewhere. Your house will have a wire with at least 3 wires in it, a black, white, and bare copper. For a 240volt you should use 12/3 to supply the electricty and it will have 4 wires in it: black, red, white, and bare copper. The black and the red supply the juice and the white is the neutral with the bare the ground.
Not sure what you mean by cutting the one wire to hook it up to the two legs of the 240. Do you mean to splice a pigtail to the one wire you have, giving you 2 and then hooking those up to the heater?
Whatever you do, it sounds like a wire is broken off somewhere. From looking at the pic, tothe left of the black wire is asmall tan nub that looks like the paper that typically wraps a wire. Is that another wire, just broken off so you can't see it?
Also the plug I see is also NOT a 240 volt plug. They look different.
Not much help I'm afraid!
Good luck, not sure the folding option will help enough, lol