This paper deals with the singular nonlinear third-order periodic boundary value problem u′″ + ρau = f(t,u), 0 ≤ t ≤ 2π, with u(i)(0) = u(i)(2π), i = 0, 1, 2, where ρ∈ (0, 1/√3) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2π] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.