Careless Singing Costs Lives

[Elmo]

In Dwimordene, in Lorien
Seldom have walked the feet of men,
Few mortal eyes have seen the light
That lies there ever, long and bright.
Galadriel! Galadriel!
Clear is the water of your well;
White is the star in your white hand;
Unmarred, unstained is leaf and land
In Dwimordene, in Lorien
More fair than thoughts of Mortal Men.

Nice version of it sung http://uk.youtube.com/watch?v=onC-hCcYtsg

My point is that should Gandalf really be singing about the star in Galadriel's right hand right in front of Wormtongue? Wormtongue would tell Saruman every word of the encounter and Saruman (clever bloke he is) would work out what the star was. And Sauron could probably garner the information from Saruman using the palantir. Some uncharacteristic foolishness from Gandalf brought about by his annoyance of Grima perhaps?

I've just realised Saruman could have already known where the Elven Ring was, but then wouldn't Sauron know where it was?