Zero conditional mean and zero correlation $\newcommand{\Cov}{\mathrm{Cov}}$
$\newcommand{\E}{\mathrm{E}}$
Is $E(Y|X)=0$ equivalent to $\Cov(X,Y)=0$?
I know $E(Y|X)=0$ implies $\Cov(X,Y)=0$, because 
$\Cov(X,Y) = \E(XY) - \E(X)\E(Y) = \E[\E(XY|X)]-E[X]E[E(Y|X)]=0$
But is the other way around true? Does $\Cov(X,Y)=0$ imply $E(Y|X)=0$?
Can anyone give a proof or disproof? Thanks.

No.
Let $Y=c$ a.s. with $c\neq0$. 
Then $\mathsf{Cov}(X,Y)=0$ but $\mathsf E(Y\mid X)=c\neq0$.
