Let \(M\) be the set of all finite complex-valued Borel measures \(\mu\not\equiv0\) on \(\mathbf R\). Set \(l(\mu)=\inf(\text{supp}\,\mu)\). The classical Titchmarsh convolution theorem claims that if measures \(\mu_1, \mu_2,\dots,\mu_n\) belong to \(M\) and satisfy \(l(\mu_j)>-\infty\), \(j=1,2,\dots,n,\) then \[ l(\mu_1\ast\mu_2\ast\dots\ast\mu_n)=l(\mu_1)+l(\mu_2)+\dots+(\mu_n),\tag{1} \] where \(`\ast'\) denotes the operation of convolution. The authors prove that if \(\mu_1,\mu_2,\dots, \mu_{n-1}\in M\), \(n\geq3\), are linearly independent over \(\mathbf C\), satisfy the conditions: \[ | \mu_j| ((-\infty,x))=O(\exp(-c| x| )),\quad x\to-\infty,\quad\text{for all}\quad c>0,\tag{2} \] and \(\mu_n=\mu_1+\mu_2+\dots+\mu_{n-1}\), then (1) remains true. Condition (2) is sharp: the statement ceases to be true if `for all' in (2) is replaced with `there exists'. This theorem is derived from the following factorization theorem in the class \(H^\infty(\mathbf C_+)\) of functions analytic and bounded in the upper half-plane \(\mathbf C_+\): Let \(h\not\equiv0\) belong to \(H^\infty(\mathbf C_+)\). Suppose that \(h=g_1\cdot g_2\cdot \dots\cdot g_n\), where the functions \(g_j\), \(j=1,2,\dots,n\), \(n\geq3\), are analytic in \(\mathbf C_+\) and satisfy the conditions: (i) there exists \(H>0\) such that \(\sup\{\sum_{j=1}^n| g_j(z)| :00\), such that \(\sup_{0