Let $(M^2,g(t))$ be a complete ancient gradient Ricci soliton with bounded positive curvature. If the soliton is compact and shrinking, then the time slices have constant positive curvature, so the model is the round shrinking sphere or the constant-curvature quotient $\mathbb{RP}^2$ allowed by the topology. If the soliton is noncompact and steady with bounded positive curvature, then the model is Hamilton's cigar soliton up to scaling and pullback by diffeomorphisms.