The spectrum of the Cesàro operator C is determined on the spaces which arises as intersections A p α+ (resp. unions A p α−) of Bergman spaces A p α of order 1 < p < ∞ induced by standard radial weights (1 − |z|) α , for 0 < α < ∞. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces A p α , with respect to α. Proving that these spaces admit the monomials as a Schauder basis paves the way for using Grothendieck-Pietsch criterion to deduce that we end up with a non-nuclear Fréchet-Schwartz space (resp. a non-nuclear (DFS)-space). We show that C is always continuous, while it fails to be compact or to have bounded inverse on A p α+ and A p α− .