Let \(B_n\) be the unit ball in \(\mathbb{C}^n\) and \(\mathcal{D}\) the Dirichlet space, that is, the subspace of analytic functions in the Sobolev space with the norm \[ \left[\sum_{i=1}^n\int_{B_n}\left(\left|\frac{\partial f}{\partial z_i}(z)^2+ \frac{\partial f}{\partial \overline{z_i}}(z)^2 \right|\right) dv\right]^\frac{1}{2}. \] The authors show that Toeplitz operators defined as \[ T_\mu(f)(\omega)=\int_{B_n}f(z)\overline{K(z,\omega)} d\mu(z) \] where \(\mu\) is a finite measure on \(B_n\) and \[ K(z,\omega)=\sum_{\alpha\in Z^n}\frac{(|\alpha|+n-1)!}{|\alpha|n!\alpha!}z^{\alpha}\overline\omega^{\alpha} \] is the reproducing kernel of \(\mathcal{D}\), form a dense set in the space of all bounded linear operators in \(\mathcal{D}\) in the strong operator topology. This is obtained as a consequence of the result stating that the closure of the set of Toeplitz operators in the operator norm topology is the space of all compact operators on \(\mathcal{D}.\)