Let M be the set of all finite complex-valued Borel measures mu not equivalent to 0 on R. Set l(mu) = inf(supp mu). The classical Titchmarsh convolution theorem claims that if: (i) mu(j) epsilon M, (ii) e(mu(j)) > -infinity, j = 1,..., n, then l(mu(1)) +...+ l(mu(n)) = l(mu(1) *...* mu(n)) The condition (ii) cannot be omitted. In 80's, it had been shown that (ii) can be replaced with sufficiently rapid decay of the measures mu(j) at -infinity and the best possible condition of this form had been found. We show that the last condition can be weakened if we dealing with linearly dependent measures mu(j) and find the best possible condition in this case. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.