We study Ricci solitons on locally conformally flat hypersurfaces Mn in space forms M˜n+1(c) of constant sectional curvature c with potential vector field a principal curvature eigenvector of multiplicity one. We show that in Euclidean space, Mn is a hypersurface of revolution given in terms of a solution of some non-linear ODE. Hence there exists infinitely many mutually non-congruent Ricci solitons of this type. Furthermore when c≥0 and Mn is complete, the Ricci soliton is gradient and in the case it is shrinking, Mn must be the product of the real line and the (n−1)-sphere.