In this paper, we investigated a boundary value problem with the Sturm-Liouville type conditions for a linear ordinary differential equation of fractional order with delay. The condition for the unique solvability of the problem is obtained in the form △= 0. The Green function of the problem, in terms of which the solution of the boundary value problem under study is written out, is constructed. The existence and uniqueness theorem for the solution of the problem is proved. It is also showed that in the case when the condition of unique solvability is violated, i.e △ = 0, then the solution of the boundary value problem is not unique. Using the notation of the generalized Mittag-Leffler function via the generalized Wright function, we also studied the properties of the function △ as λ → ∞ and λ → −∞. Using asymptotic formulas for the generalized Wright function, a theorem on the finiteness of the number of eigenvalues of a boundary value problem with the Sturm-Liouville type conditions is proved.