Regularity is proved for an arbitrary generalized solution of a quasilinear elliptic equation of divergent type which belongs to $W_2^{m+n/2}(\Omega')$ , for an arbitrary strictly interior subregion $\Omega'$ of a region $\Omega$ ( $2m$ is the order of the equation, and $n$ is the number of arguments). It follows from this, in particular, that the regularity problem has an affirmative solution in the two-dimensional case.