In this paper, the singular three-point boundary value problem { u ″ ( t ) + h ( t ) f ( t , u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = α u ( η ) , where η ∈ ( 0 , 1 ) , 0 < α < 1 / η , is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator. h ( t ) is allowed to be singular at t = 0 , 1 and f may be singular at u = 0 . The existence of a positive solution is obtained by a fixed index point.