The authors investigate the first boundary value problem for the stationary Stokes system in an exterior domain. Let \(\Omega\) be an exterior domain in \({\mathbb{R}}^ n\) (n\(\geq 3)\) with the smooth boundary \(\Gamma\). The following problem is considered in Sobolev spaces: find the vector \(u(x)=(u_ 1,...,u_ n)\) and the scalar function p(x) satisfying the Stokes equations \[ -\Delta u+\nabla p=f,\quad div u=g\text{ in } \Omega \] and the boundary condition \(u|_{\Gamma}=\phi\). The existence and uniqueness of a weak solution \((\nabla u\in L_ q\), \(p\in L_ q)\) are proved. A priori estimates of this solution are established. The existence and uniqueness of a strong solution \((D^ 2u\in H^{m,q}\), \(\nabla p\in H^{m,q})\) are proved, too. Precise estimates of a strong solution are obtained. Proofs are based on methods of functional analysis.