I have seen pictures of candles burning in space, and the flame was completely blue. This means that the temperature of the flame is actually higher than on earth, contrary to what bucko says. The amount of wax being burned, and hence the amount of energy released is of course lower, as the supply of oxygen to the flame depends on diffusion rather than convection, as convection is driven by gravity acting on a system with hot and cold gasses. But because convection also doesn't bring in any fresh, cool air to the flame it can reach higher temperatures. This leaves us with a hot but very small flame, as a small flame would also maximise the surface-to-volume ratio of the flame, which is advantageous because the surface determines how much oxygen can enter the flame and the volume determines how much oxygen is needed for the combustion taking place. Also, a smaller flame decreases the distance oxygen has to penetrate into the flame. In the absence of macroscopic mixing of fuel and oxygen, all these factors suddenly become relevant.
Also, as the length of the wick underneath the flame is an interplay of forces, in an environment with gravity the flame creeps up the wick because hot gasses rise. Now this is counterbalanced by the necessary supply of fuel to the flame, which will stop if the flame doesn't melt enough wax. The flame will dwindle, first consuming all the fuel higher up in the wick, gradually creeping down the wick, until it starts melting enough wax to feed itself again. This tendency of the flame to "go up" of course wouldn't play any role in an environment without gravity, so the flame would in all probability touch the surface of the candle itself.
Now there is one difference between the "bottom" (the end extending into the candle) of the wick and the "top" (far end) of the wick: the bottom of the wick is connected to the reservoir of molten wax. As the flame also burns farther along the wick, this means that there is molten wax present in the wick at the "bottom" end of the flame. This will burn before the material of the wick burns. Now at the far end the flame has drained the wick of fuel entirely, and thus the wick itself will burn. This has nothing to do with the edge of the flame being the hottest part. On the contrary, I would suppose that the edge is the coolest, also evidenced by the blue color of the inner and the yellow color of the outer parts of the flame of a candle burning on earth.
Now this serves to show that such a simple experiment actually is very complex matter, with an interplay of many, many forces, and I wouldn't be surprised to hear that the reason they actually did this experiment was to settle some very heated discussions.