Pick 2 spare pieces of mdf/plywood. In the first of about 20x20 cm you drill a hole of around 20mm in the middle. Thats where the router bit goes through and will be on top of the sheet that needs to be cut. Next you put the second piece under the sheet where the hole comes (it will hold the loose circle after you went round, and makes sure you don't route into the table. If the table may get little damage, just use the table as second piece!
Then screw a screw on a certain distance from the hole in the first piece, through the circle center in the actual sheet that needs to be cut, and through the help sheet underneath (or into the table). The distance can be calculated, but it is better to make a test circle about the final size in spare wood, and then measure the difference in diameters to adjust the distance of the screw to the routerbit-hole.
Then you screw the router on the 20x20 cm help sheet, centering the routerbit with the hole in the middle. Finally fasten the actual sheet and the bottom help sheets with clamps onto the table.
Now you can start routing with the around-the-screw rotating helpsheet. Go 3mm deeper every turn until you get through. Because the screw holds the cutaway part, it won't fly away or kill the routerbit by locking it up.
If i was not on holiday, i would have photoshopped a sketch.