The disc is either snap-on or screw-on, as HJ indicates. (HJ is the forum's "plumber's plumber", a brilliant guy with encyclopedic knowledge -- when he gives you tips you can take them to the bank, so heed his suggestions.)
Here's a universal disc that will replace either type, available at Lowe's:
http://www.lowes.com/pd_330407-143-...ntURL=?Ntt=plumb+pak+toilet&page=2&facetInfo=
There is a similar Danco one at HD, but the reviews indicate that it may not work as well as the Plumb Pak one at Lowes.
Your toilet looks like a one-piece, so installing a modern flapper flush valve is a little more involved, in terms of aquiring the parts and installing them, than just going to Lowe's and buying a Korky flush valve kit, which you could do with a two-piece toilet. Best just to try (carefully) to fix this one with a new disc.
Korky also makes a replacement disc, Model No. 421BP:
http://www.korky.com/index.php?p=Product&i=1001#.U5jMUvldUeg It is likely a little harder to find locally than that Plumb Pak at Lowe's. Look at the link because the instructions there may be helpful regardless of brand (even if they are a bit schematic).
PS The way that tilt-flush works is that there is a weight in the bigger piece of tube; when you pull on the handle, the chain lifts the assembly vertical, and water pours out of the tank into the flush valve opening. What you thought was the overflow riser is actually a piece that, when full of water, counterbalances the weight. Water runs out a hole in the bottom of it as the level in the tank falls, and when the water in the tube (which is holding the valve open) gets down to a certain level, it no longer counterbalances the weight and the valve flops closed and positions that disc over the opening, thus sealing it. Not too long ago, I replaced the tilt-flush with a regular flapper system in one of our toilets, which had come with the tilt-flush when it was installed in the 1970s (maybe even the late 60s).