This Letter presents the first experimental demonstration of the capability to launch shocks of several-hundred Mbar in spherical targets--a milestone for shock ignition [R. Betti et al., Phys. Rev. Lett. 98, 155001 (2007)]. Using the temporal delay between the launching of the strong shock at the outer surface of the spherical target and the time when the shock converges at the center, the shock-launching pressure can be inferred using radiation-hydrodynamic simulations. Peak ablation pressures exceeding 300 Mbar are inferred at absorbed laser intensities of ∼3×10(15) W/cm2. The shock strength is shown to be significantly enhanced by the coupling of suprathermal electrons with a total converted energy of up to 8% of the incident laser energy. At the end of the laser pulse, the shock pressure is estimated to exceed ∼1 Gbar because of convergence effects.