Linear programming IFF with equality constrain

Question

Is it possible to write the following logical constrain in linear programming?

Let $v$ be an integer variable and $k$ an integer constant. Let $y$ be a binary variable. The logical constraint is

$y=1 \Longleftrightarrow v=k$.

I need this kind of constraint in linear programming to use it in AMPL, but I really can't find a way to write it down as a linear constraint.