Rear-wheel drive conversions are nothing new--enthusiasts have been fighting the Evil Transaxle Empire since well before it came to dominate the highways. The standard rear-drive procedure is to cut the car into bits, fabricate a new chassis and reassemble the thing around a properly-oriented engine and drive train. However, there is a vastly simpler and cheaper way that will yield you so much more than a mere front engine/rear-drive set-up--and you may only need the car down for a weekend.
Buy a running, automatic transmission, front-wheel drive donor car with the engine you want to use. If you're going to go through the trouble of converting the car, you might as well use the engine you really want for a huge bump in power. If converting a four cylinder, pick a V6 model from the same manufacturer. Examples: a V6 Acura RL donor for a Honda Civic, a Lexus ES300 donor for a Camry or a Nissan Altima donor for a Sentra. Cadillac's Northstar and Ford's modular 4.6L are both excellent choices if you're looking for a V8 front-drive donor.
Cut the entire front clip off of the donor car, including the engine, transmission, suspension, front cross-member and strut towers (if it has McPherson strut suspension). Do not damage the computer or any of the wires.
Create a subframe for the drive assembly by welding tubular supports along the corners of the strut towers and connecting them to the frame. Then connect the strut towers with a removable brace that bolts to the towers' strut mounts.
Install a fixed toe-adjustment bar designed for any rear-drive car with independent rear suspension (such as the Chevrolet Corvette, Toyota Supra or some Ford Mustangs) in place of the stock power steering rack. Try to locate a toe-bar that will thread into the stock steering end links.
Cut the rear of your car apart so that the assembly physically fits in the car's boot and rear seat area. Connect the welded subframe to the car's body by whatever means best suits your application. Quarter-inch plate steel works well as does rectangular tubing. This is a highly custom fit so enlist the help of an experienced fabricator if this is your first time with this kind of project.
Fabricate a lightweight frame out of square stock tubing around the engine and cover the framework with 1/2-inch pressure-treated plywood. Connect the plywood to the frame with piano hinges for easy engine access. Although plywood may seem somewhat low-tech for this application, the fact is that wood is an excellent insulator against both heat and noise penetration. Besides, no one will know what it is once you cover the wood with upholstery on the inside and thermal reflective material on the engine side.
Remove the car's stock engine and transmission and install the donor car's fuel tank in their place.
Re-connect the new engine to the donor car's computer, fuel tank, A/C lines and condenser, radiator, battery and required gauges. Connect your car's shifter to the new transmission with either a universal shifter cable or by simply extending the wires that connect the transmission to the shifter sensor. In theory, your new mid-engined, rear-drive monster should fire up and run just as well as the donor car.
When choosing a donor, measure to ensure that your car's rear quarter panels will be wide enough to fit the donor's strut towers between them. You're going to lose your back seat and a good part of your boot, but it's worth it. For instance, a well engineered 1995 Honda Civic (about 998kg.) with a 300-horsepower Acura RL V6 would have about the same power-to-weight ratio as a Nissan GT-R and handling on par with a Honda NSX, if not a Lotus Exige.