IF you're goal is lightweight. . .then find something else.
No car even comes close to as light as you can get it, except maybe the
citroen 2CV.
If you are honestly out for lightweight. . .Find a small aluminum chassis(think Go-kartesque)(no more than 500 lbs) with space for a car sized engine. Find the smallest diesel you can and on a wrecked car and buy the engine trans CV joints and steering assemblies. Put the engine in the back and tool the steering if you need to(the chassis might come with it) for the front.
You'll have a vehicle at under 1000#s with plenty of power and you can cruise at idle around 30-45 mph. coroplast, fiberglass, whatever as a body and you could easily do 80-90 mpg with a passenger and tons of play if you wanted it.
existing cars are very heavy because they are designed to be ridden, not save fuel. If the chassis weighs more than the engine you're kidding yourself that its all about the weight. A small kubota with revamped pistons and rods with a turbo could do all the power you'd need and its only going to weigh. . .maybe 200 lbs.
An aluminum(if you are loaded and can pay someone to manufacture a carbon fiber or fiberglass monocoque platform do that) chassis with no body carbon fiber racing seats steering and wheels ought to be under 400-500 lbs.