We report the first integrated experiments of the double-cone ignition (DCI) scheme, conducted on the ShenGuang II Upgrade (SG-II-U) facility using 14 kJ nanosecond (ns) laser beams and a 450 J picosecond (ps) laser beam. Two carbon-deuterium (CD) shells were imploded by the ns laser beams inside dual compressing-cones. Heated by the ps laser beam to 600 ± 50 eV, we achieved a heating efficiency of 15.6 ± 6.2% and a consistent neutron yield of up to (6.0±0.5)×10^5, with a shot-to-shot fluctuation of only 7% across all three shots at optimal ps-laser timing delay. Multi-scale simulations reveal that this high ps-laser-to-fuel heating efficiency originates from an extremely sharp compressed plasma density gradient with a simulated scalelength of ~8 μm at the edge of the high-density plateau, a unique geometric benefit of the dual-cone DCI architecture. The steep edge along the heating direction enables the heating-cone to be placed as close as 35 μm to the plasma center while preserving its structural integrity throughout ps-laser heating. This minimized separation mitigates energy losses caused by large divergence of relativistic electron beams (REB) generated by the ps laser beam. We predict that heating efficiency can be further elevated to 20%-30% by scaling the laser energies to the levels of our next-generation high-energy laser facilities currently under construction.