Theoretical study of the chiral liquid crystal (CLC) director distribution in a wedge shape cell with weak surface anchoring is presented. It is found that for a sufficiently short pitch CLC the well known defect lines separating the wedge area differing by the number of director half turns N at the wedge thickness may be replaced by nonsingular walls. It should be noted that the term "weak surface anchoring" really is related to the large values of the dimensionless parameter S-d = K-22/Wd, where K-22 is the elastic twist modulus, d is the layer thickness and W is the depth of the surface anchoring potential. So, at any strength of the anchoring a sufficiently thin layer (small d) insures the conditions of "weak surface anchoring". At sufficiently thick area of the wedge the well known picture of the defect lines separating the wedge areas differing by the number of director half turns restores. The calculations of the director distribution in a wedge shape cell with infinitely strong anchoring at one surface and finite anchoring strength at the second one performed for the two sets of model anchoring potentials reveal qualitative difference in the director distributions for the Rapini-Papoular-like and B-like model surface anchoring potentials. The results show that the experimentally distinguishable details of the director distribution in the wedge area with nonsingular walls allow one to obtain information on the shape of the surface anchoring potential and estimate the energy of the defect lines replacing the nonsingular walls.