The effect of spin splitting caused by structural asymmetry (Rashba's contribution) and bulk asymmetry (Dresselhaus's contribution) on the magnetoconductance of two-dimensional structures with high mobility of charge carriers is studied. The theory of weak localization with regard to both of the contributions is developed. The theory is valid in the entire region of classically low magnetic fields for arbitrary relations between the frequencies of spin precession and elastic collisions. The suppression of the correction for antilocalization is demonstrated in the case of equal contributions of structural anisotropy and bulk anisotropy to the spin splitting. The effect of the contribution, cubic in the wave vector, to the spin splitting on the quantum magnetoresistance is studied.