The elasticity problem for a homogeneous elastic medium containing a stiff axisymmetric fiber is considered. The solution depends on two small parameters: the ratio of the transverse size of the fiber and its length, and the ratio of the elastic moduli of the host medium and fiber materials. Construction of the principal terms of the elastic field in the medium with a fiber is reduced to a differential equation for the axial component of the stress field inside the fiber. The asymptotic solution for cylindrical, ellipsoidal, and double cone files is presented.