Though I have come to be very fond of antler axes, I like stone axes too. The advantages with ground stone over antler is, though not much sharper, it holds it's edge better. In comparison to flaked axes the edge is more stable and unlikely to chip unless you hit a stone or something.
I have made two of these axe (celt) heads before. One was hafted and tested and it definately worked very well. The second one still needs to be hafted. On this photo that particular head is found on the far right, the pecking stone (previously a hand axe from quartzite) to the left and the head in progress in the middle.
River cobbles often provide a minimum of work to make an axe. The one in centre on this photo has been integrity tested by chipping off a little piece in the front. Good axe stone is homogenous and hard.
The first stage in producing an axe head is to peck it down to the approximate shape. Be careful on the edge, not to chip the stone. The sharper a piece you use as a pecking stone, the more precise you can do the pecking and it is also faster. The next phase on making this axe is to grind the edge of the head sharp. By the way, I don't usually peck stone in the bedroom, this photo is staged. ;-)