The author studies regularity properties of the weak solutions of the Dirichlet problem \[ - {\partial \over {\partial x_ i}} \biggl( a_{ij} {{\partial u} \over {\partial x_ i}} \biggr)- {\partial \over {\partial x_ i}} (b_ i u)= {{\partial f_ i} \over {\partial x_ i}} \quad \text{in } \Omega, \qquad u=0 \quad\text{on } \partial\Omega, \tag{1} \] if \(a_{ij}\in L^ \infty (\Omega)\), \(a_{ij}= a_{ji}\), the principal part of the equation is uniformly elliptic, \(b_ i\) and \(f_ i\) belong to some Morrey spaces. The novelty of the regularity results of the paper is the fact that the weak solution of the Dirichlet problem (1) can have certain regularity properties (belonging to some Morrey spaces or Hölder continuity) even in the case when the functions \(b_ i\) and \(f_ i\) do not possess high integrability requirements \((b_ i, f_ i\in L^ p\), \(p\in {1\over 2}n\) or \(p>n)\).