Jack has the best idea..use the old stuff to mule-in the new stuff.
You got about an 80% chance that the coax is stapled into place....probably won't work...give it a light tug, if it doesn't feel loose, give up.
If the coax run is not too long, you might be able to replace or adapt the "F" connectors to BNC, then use a 10BASE-2 setup. The problem is that the RG59 is 75 ohm cable, and the Coax used for Ethernet is 50 ohm...a significant impedence mismatch. The usual cable used resembles RG58 (50ohm), and was designated as "Ethernet Thin Coax"...a lot of installations went in with Radio Shack RG58 and worked OK...but there is actually a specific cable type for this kind of network.
Once upon a time, 3COM made some impedence matching devices called "Pair Tamers" that allowed you to connect Ethernet up to 75 ohm cable (they had some for 92 ohm stuff too). If you scan around (google? Ebay?) you might be able to find some. They used to be ~US$50.00 a piece (10-15years ago)...you could probably get 'em for 5-10 a piece now.
In case you're not familiar with coax Ethernet, you have to put a BNC "T" connector on the back of the NIC, one branch of the "T" goes to the network, the other branch of the "T" is either terminated with a terminator...or continues to the next PC. The last two devices on the cable should have a terminator on one branch of the "T."
You CANNOT run the cable directly into the NIC, you must use the "T" and terminators.
All-in-all, I'd say you have better than 50% chance of getting it to work, maybe more.
Good Luck, hopefully the cable Gods will cut you some slack on this one......
Scott