The multiplicity features and the secondary flow structure of the fully developed, laminar flow of a Newtonian fluid in a straight pipe that is rotating about an axis perpendicular to the pipe axis are examined. The governing equations of motion are solved numerically using the control volume method and the SIMPLE algorithm. The solution structure is governed by two dynamical parameters, Ekman number, Ek=ν/D2Ω and Rossby number, Ro=U/DΩ, where D is the pipe diameter, ν is kinematic viscosity, Ω is rotational speed, and U is velocity scale. Results are presented for a fixed Ekman number of Ek=0.01 and a range of Rossby numbers between 0 to 20. The primary solution branch begins as a unique solution at low Rossby numbers. Its secondary flow structure consists of two-cells. At higher values of Ro a hitherto unknown solution with a four-cell flow structure appears, which coexists with the two-cell flow structure over a range of Ro up to 20. Transient, two-dimensional simulations were carried out to determine the stability of the solutions to two-dimensional perturbations. The two-cell flow structure is stable to both symmetric and asymmetric perturbations. Four-cell flow structure is stable to symmetric perturbations and unstable to asymmetric perturbations, where it breaks down to a two-cell flow structure.