Is it safe to use a log burning stove in an old fireplace?
The way fire insert stoves work is relatively simple in conversion. 

You cut the damper out of the fireplace, run stainless steel pipe (6" dia.) up the inside of the chimney. The lengths are held together with 3-6 sheetmetal screws so this is an actual inside pipe assembly and it hangs off a sheetmetal cap that covers the top of the chimney and is silicone caulk sealed to the top of the chimney flue tile so it's air tight to prevent exactly the type of heat loss you mention. The pipe that extends above the sheetmetal sealing cap gets a standard stovepipe cap to prevent downdrafts and water intrusion from precipitation.

The bottom gets a piece of flex stainless pipe that joins the chimney stack to the stove.

The advantage of this is you are no longer at the mercy of cracked flue tiles  and the fire hazard that represents. The 6 inch pipe runs at a higher temperature and tends to not creosote up. There are standard round chimney brushes for these so the soot that accumulates near the top is easily removable by popping the cap off and running the brush and 3-4 rods down the length of pipe.

Depending on where you live and your county's rules, this conversion may require a permit, but it saves you a lot of lost heat and drastically lowers your liability from flue fires. 

The stainless pipe will need inspection periodically for pinholes and monitoring for soot buildup and a good brushdown every so often.