Anton, I think you're getting confused here.
The sync speed you can set on the camera is actually the *maximum* sync speed. But it has nothing to do with the shutter speed that is used for any shot.
In Av mode, the camera will set a shutter speed that is appropriate for the subject and the ambient lighting, regardless of whether you use flash. If you do then use flash, the metering system will add some flash as a fill-in, but the shutter speed is *not* changed by the use of flash. It will still be too slow.
If you want to use Av mode, tough. That's the way it works.
If you want to use the flash as the sole source of illumination, switch to M mode and dial in an exposure like 1/100th at f/8. Then the ambient light will contribute virtually nothing to the exposure and the flash will provide all the illumination.
Of course there are other possibilities too; for example in M mode you can set the exposure to be say 1-2 stops below what is required by the ambient lighting, so that the areas not reached by the flash (e.g. shadows!) aren't so dark. But the key to it all is to remember that - in Canon SLRs at least; I believe Nikons behave differently - the camera's meter does its job independently of whether or not you use flash.