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Well it's not wet, but if you stopped the reactions, the sheer force of gravity of the upper layers on the inner layers would start them again. Not that that has any effect on its mass. (I mean it does, because the nuclear reactions convert mass to energy, but that's a very long process.)
Asking weight doesn't really make sense, because weight is a product of gravity. The sun has a mass of 1.9891x10^30 kg, and at 1g that's 4.384x1030 lb, at least according to the Google result summary I copy-pasted from.
How much would the earth weigh if it was sitting on the frozen surface of the switched off Sun?
I'm sure it would be the same as it currently does.
Weight depends on location. For example, you would weigh less on the moon.