normal random variable with zero variance

I have used the theorm of normal correlation which states the for a jointly gausian pair the conditional distribution of 1 of the random variables is again gausian with mean E[X(0)/X(1), and variance given by cov{X(0)/X(1)

I have calculated this and obtained E[X(0)/X(1),=X(1) and cov{X(0)/X(1)=0. where X(0), X(1) are uncorelated N(0,1. ) My teaher said this was corect but i cant figure out what the implcations are. Does this mean that the coresponding distribution will not exist, or can it actually have zero variance. Or does imply that the random varibale is not random but rather deterministic conditionally and therefore has no distribution?