Let[Figure not available: see fulltext.] be the collection of parallelepipeds in R^κ with edges parallel with the coordinate axes and let[Figure not available: see fulltext.] be the collection of closed sets in R^κ. Let π(G, H)=inf {ε{divides}G{A}≤H{A^ε}+ε, H{A}≤G{A^ε}+ε for any[Figure not available: see fulltext.]; L(G, H)= inf {ε{divides}G{A}≤H{A^ε}+ε, H{A}≤G{A^ε}+ε for any[Figure not available: see fulltext.], where G, H are distributions in {Mathematical expression}. In the paper one gives the proofs of results announced earlier by the author (Dokl. Akad. Nauk SSSR, 253, No. 2, 277-279 (1980)). One considers the problem of the approximation of the distributions of sums of independent random vectors with the aid of infinitely divisible distributions. One obtains estimates for the distances π(·, ·), L(·, ·) and[Figure not available: see fulltext.]. It is proved that[Figure not available: see fulltext.], where 0≤p_i≤1, {Mathematical expression}; E is the distribution concentrated at zero; V_i(i=1, ..., n) are arbitrary distributions; the products and the exponentials are understood in the sense of convolution.