The origin and distribution of stellar-mass black hole spins are a rare window into the progenitor stars and supernova events that create them. Swift J1728.9-3613 is an X-ray binary, likely associated with the supernova remnant (SNR) G351.9-0.9. An NuSTAR X-ray spectrum of this source during its 2019 outburst reveals reflection from an accretion disk extending to the innermost stable circular orbit. Modeling of the relativistic Doppler shifts and gravitational redshifts imprinted on the spectrum measures a dimensionless spin parameter of a=0.86+/-0.02 (1{sigma} confidence), a small inclination angle of the inner accretion disk {theta}<10{deg}, and a subsolar iron abundance in the disk AFe<0.84. This high spin value rules out a neutron star primary at the 5{sigma} level of confidence. If the black hole is located in a still visible SNR, it must be young. Therefore, we place a lower limit on the natal black hole spin of a>0.82, concluding that the black hole must have formed with a high spin. This demonstrates that black hole formation channels that leave an SNR, and those that do not (e.g., Cyg X-1), can both lead to high natal spin with no requirement for subsequent accretion within the binary system. Emerging disparities between the population of high-spin black holes in X-ray binaries and the low-spin black holes that merge in gravitational wave events may therefore be explained in terms of different stellar conditions prior to collapse, rather than different environmental factors after formation.