Year upon year you dig down, until eventually you have another floor's worth of space, then the original house becomes the top/second floor, and the new "basement" becomes a new area.
I would add that I wouldn't want soil to be literally the walls of the lower level, as frozen soil conducts away heat from the air extremely effectively. So what you'd likely want to do is set up an insulated layer between the soil and the living space.
It is also worth noting that heat rises, so you'll still lose a lot of heat if the new basement doesn't have a VERY well insulated roof. Then you could just pop out the front of the old house, add doors, and now you have a garage for the car.
It isn't that crazy of an idea. But it would take several years (minimum 3-4) and cost you a few thousand just for insolation (both new roof and new walls), and also for new structural supports if you want to do the house-to-garage thing (to hold the car's weight).
That's what documents like http://www.uaf.edu/files/ces/publications-db/catalog/eeh/HCM... say:
> Make every attempt to keep the soil beneath the building frozen and the permafrost stable. The strategy for alleviating the engineering risks of building on permafrost sites is to build the structure on piles or an elevated foundation, taking special care to insulate the ground and prevent thawing.
Permafrost melts. Houses in a swamp. \o/ http://www.epa.gov/climatestudents/impacts/signs/permafrost....