In this Letter, we show that multiband observations of stellar-mass binary black holes by the next generation of ground-based observatories (3G) and the space-based Laser Interferometer Space Antenna (LISA) would facilitate a comprehensive test of general relativity by simultaneously measuring all the post-Newtonian coefficients. Multiband observations would measure most of the known post-Newtonian phasing coefficients to an accuracy below a few percent-2 orders-of-magnitude better than the best bounds achievable from even "golden" binaries in the 3G or LISA bands. Such multiparameter bounds would play a pivotal role in constraining the parameter space of modified theories of gravity beyond general relativity.