We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix $H{ₙ}^{\left(0\right)}$ and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of $H{ₙ}^{\left(0\right)}$ and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...