Prepare a game board with as many squares or steps as there are lines or phrases in the song, plus 2. Place some kind of marker at the "start" and a picture or keyword that describes the message of the song at the "finish."
Have the children listen to you sing the first phrase. Then invite them to sing it with you. Next they should sing it by themselves. If they can sing it well by themselves they can advance the marker one step. If they cannot, repeat the process. Do this with each phrase until you reach the final two squares. Ask the children to sing the entire song twice to advance to the finish.
The picture shows a simple game board that I drew on the chalkboard to use with the song "The Golden Plates." The pictures come from the Primary Visual Aid Cut-outs (Set 7: Pioneers and Restoration).
P.S. It occurs to me that I should mention something more about this game. You can see that it isn't competitive in any way. I don't much care for competition in primary (or anywhere else, for that matter) so, depending on how considerate your primary kids are, and how "trained" they are at loving competitive games, they may wonder what the point of this game is. The point is simply to get the marker to the finish line. That may not be enough competition for some kids. If that is the case, they will find the game boring. This is a simple game that even my younger kids can enjoy. I have a combined group and most of them are willing to play a game that is played just to reach an objective. Even the older ones in my group are considerate enough to not make comments about how something is boring or babyish, although sometimes I catch them looking at each other and rolling their eyes ;o) If this isn't the case with your group, I suggest that you try to retrain them so that you can enjoy something like this, or choose something different.