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Using a combination of Hubble Space Telescope and James Webb Space Telescope imaging, a runaway supermassive black hole was recently identified with an inferred velocity of <inline-formula><mml:math><mml:mrow><mml:msubsup><mml:mrow><mml:mn>954</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>126</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>110</mml:mn></mml:mrow></mml:msubsup><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:mi>km</mml:mi><mml:mtext> </mml:mtext><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, likely ejected from a compact star-forming galaxy at <inline-formula><mml:math><mml:mi>z</mml:mi><mml:mo>≍</mml:mo><mml:mn>0.96</mml:mn></mml:math></inline-formula>. Assuming the runaway black hole originated from a gravitational-wave-driven merger of two supermassive black holes (SMBHs), we combine its measured recoil velocity with gravitational-wave recoil predictions from numerical relativity and black-hole perturbation theory to constrain the mass ratio and spin configuration of the progenitor binary that overcame the final-parsec problem and merged <inline-formula><mml:math><mml:mo>∼</mml:mo><mml:mn>70</mml:mn><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:mi>Myr</mml:mi></mml:math></inline-formula> ago. We find that the progenitor binary must have been precessing, with a mass ratio <inline-formula><mml:math><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>≲</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>, and that the more massive SMBH likely possessed a high dimensionless spin magnitude (<inline-formula><mml:math><mml:mo>∼</mml:mo><mml:mn>0.75</mml:mn></mml:math></inline-formula>) in order to generate a recoil of this magnitude. Such SMBH mergers could represent an interesting source population for the upcoming Laser Interferometer Space Antenna mission, with characteristic signal-to-noise ratios of order <inline-formula><mml:math><mml:mo>≳</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>. Furthermore, the inferred progenitor SMBH properties suggest that the compact galaxy likely originated from a major, gas-rich ("wet") merger between two galaxies of comparable mass, with a mass ratio <inline-formula><mml:math><mml:mo>≲</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>.