The objective of the presented study is to perform a vibration analysis and investigate the stability of a viscoelastic-fluid conveying pipe with an intermediate support. The mathematical model is elaborated in the framework of the Euler-Bernoulli beam theory in combination with the Kelvin-Voight viscoelastic approach. The resulting differential equation of motion and the related boundary conditions and compatibility conditions in the mid-span support are solved analytically using a power series method. The results show that an intermediate support located at ξ s = 0.1 and ξ s = 0.5 increases the critical velocity up to 35% and 50.15%, respectively. Also, the non-dimensional critical velocity for an intermediate support at ξ s = 0.1 is 4.83.