  Following Sarason's classification of the densely defined multiplication
operators over the Hardy space, we classify the densely defined multipliers
over the Sobolev space, $W^{1,2}[0,1]$. In this paper we find that the
collection of such multipliers for the Sobolev space is exactly the Sobolev
space itself. This sharpens a result of Shields concerning bounded multipliers.
The densely defined multiplication operators over the subspace $W_0 = \{f \in
W^{1,2}[0,1] : f(0)=f(1)=0 \}$ are also classified. In this case the densely
defined multiplication operators can be written as a ratio of functions in
$W_0$ where the denominator is non-vanishing. This is proved using a
contructive argument.
