It’s not that the headquarters believed the Ardennes were impossible to cross. They just believed that it wasn’t possible for a sizeable force to cross them quickly following narrow roads and launch a major attack in this area without letting time for the french army to react. And once again, the french commanders din’t expect at all a “blitzkrieg”. That for instance the german armored units would just ignore and bypass pockets of resistance and go on their merry way as quickly as possible.
Also, this part of the border had not been left totally undefended. But it was defended by reserve units of poor quality and poorly equiped, and without reserves backing them.
Beside, the allied expected the main attack to occur through Belgium and were awaiting for it. The Germans did launch an attack in Belgium, the french army and the BEF rushed in as planned. Everything seemed to happen as expected, except for the fact that the German attack (though a major one) was only a diversion move and the main thrust was intended to take place in the Ardennes. When it happened, the allied forces were caught in Belgium, had an extremely hard time trying to retreat (lack of motorized unit able to move quickly, poor logistics, roads clogged by refugees, etc…) hence were of no use until they were encircled. Actually many soldiers just marched north into Belgium, never met the ennemy, were ordered to march south back to France, and found themselves encircled without having even fought.
Once the germans had bypassed the french defense lines in the Ardennes area, there was esssentially nobody to stop them (no reserves, as I mentionned, and the bulk of the army in Belgium). Beside, the allied command still didn’t know immediatly what were the german’s intents, hence how to react. It could have been :
-Going east and attack the Maginot line from the rear
-Going south to take Paris
-Going west to cut the allied forces from mainland France (which they did), which wasn’t necessarily the most likely scenario, since the commanders still didn’t think that the german forces could move quickly enough to achieve such a feat.