No, on Christmas Eve of 1942 the German public was not aware that Stalingrad had been encircled. The rational behind the decision not to inform the German public was that the German soldiers would soon break out and there was no need to worry the public. The Germans back home did know that fighting was going on around Stalingrad, and they had revived reports that minor set backs were occurring or that the fighting was lasting longer then expected, but it was always with an optimistic overtone.

The Nazis first admitted that the 6th army was encircled on January 16th, but that announcement was accompanied by assurances by the Nazis that the men in Stalingrad would be rescued and that they were bravely fighting for the safety of the Reich. Letters were sent to loved ones all through out the battle of Stalingrad, but the letters were hopeful and showed that despite the overwhelming odds, most German soldiers remained loyal towards Hitler and truly believed that they would be rescued. Because of this, the German people couldn't truly grasp how bad things had gotten in Stalingrad until the battle was over. An example letter from an NCO:

The world outside should also know what German soldiers have taken upon themselves and have been able to bear here. This heroism and this perseverance are unprecedented and will be rewarded by a great victory.

Goebbels was given the job of turning the Stalingrad debacle into something presentable to the German public. He sent specific instructions to every newspaper and radio station instructing them how to present Stalingrad. He said:

The whole of German propaganda must create a myth out of the heroism of Stalingrad which is to become one of the most treasured possessions in German history.’ The Wehrmacht communiqué, in particular, must be phrased in a way ‘which will move hearts for centuries to come

The announcement went out, 24 hours after the 6th army had surrendered, on February 3rd, 1943. The announcement said:

The supreme command of the Wehrmacht announces that the battle of Stalingrad has come to an end. True to its oath of allegiance, the Sixth Army under the exemplary leadership of Field Marshal Paulus has been annihilated by the overwhelming superiority of enemy numbers… The sacrifice of the Sixth Army was not in vain. As the bulwark of our historic European mission, it has held out against the onslaught of six Soviet armies… They died so that Germany might live

Hitler ordered state radio to play somber music and all places of entertainment were closed down. Three days of mourning were declared, but strangely enough newspapers were not censored, nor were flags lowered to half mass.

Sources:

Hitler's Army by Omer Bartov

Stalingrad by Antony Beevor

The radio linkup to Stalingrad was fake, the men "at the front of Volga" didn't really join their comrades in singing Stille Nacht. It would boost morale on the homefront, and be good for propaganda purposes to orchestrate the deception.

How much the average german civilan would've known is up for debate. That many knew seems to be the opinion of most historians. They would have known something wasn't right by christmas. The average german would not have known about the encirclement, or how truly dire the situation was. In September and Oktober german propaganda suggested that victory at Stalingrad was just around the corner, then, following the sovjet counter-offensive, nothing but ominous silence was heard. Hitler declared in a public speech (in the Berlin Sportpalast) on 30 September 1942 that the German army would never leave the city, and gave the impression that victory was close. There were enough indications of hard fighting to make it plain that things were not going according to plans. (To add to the feeling that the war wasn't going as planned, allied forces had landed in north-africa, the winter war had been horrible, bombing raids on german cities - it was obvious to many people in Germany that things were not good.)

There were 1/4 million soldiers trapped at Stalingrad, and the letters they sent back home made it clear that the vast majority knew that the situation was hopeless. They were starving, froze to death, lacked fuel, lacked ammunition and were facing a battle they could not win, and the letters they sent home said as much. The rumours of approaching disaster quickly spread, and the mood in the population sunk. It was apparant to most that the prospect of a horrible defeat was looming at Stalingrad.

The Wehrmacht report on 16 january spoke of "heroically defensive struggle against the enemy attacking from all sides". The press was instructed to speak of "the great and stirring heroic sacrifice which the troops encircled at Stalingrad are offering the German nation". This probably didn't make anyone feel more optimistic. Then, at 3. february, the news broke that the soldiers had fought to the last shot shot "so that Germany might live".

Edit: Sources; Ian Kershaw - Hitler 1936 - 1945: Nemesis, David Glantz - Barbarossa: Hitler's Invasion.

Is there a video of this broadcast?