We propose and experimentally demonstrate a broadband slow-wave system in a surface-wave photonic crystal based on a non-uniform line-defect waveguide with graded metallic pillars, whose dispersions and cutoff frequencies vary gradually along the waveguide. Since the group velocity of designer surface plasmons at the cutoff frequencies is zero, we show that surface waves can be slowed and stopped at different positions under different frequencies. Experimental evidence including transmission measurements and direct near-field profile imaging performed in the microwave frequencies validate the stopping of surface waves in a broadband range within the forbidden bandgap of the surface-wave photonic crystal. This proposal is a promising candidate for slow-wave devices implemented on a single metal surface in both the microwave and terahertz frequency ranges.