  We argue that nuclear magnetic resonance experiments are a site-sensitive
probe for the electronic spectrum in the mixed state of the high-$T_c$
cuprates. Within a spin-fermion model, we show that the Doppler-shifted
electronic spectrum arising from the circulating supercurrent changes the
low-frequency behavior of the imaginary part of the spin-susceptibility. For a
hexagonal vortex lattice, we predict that these changes lead to {\it (a)} a
unique dependence of the $^{63}$Cu spin lattice relaxation rate, $1/T_1$, on
resonance frequency, and {\it (b)} a temperature dependence of $T_1$ which
varies with frequency. We propose a nuclear quadrupole experiment to study the
effects of a uniform supercurrent on the electronic structure and predict that
$T_1$ varies with the direction of the supercurrent.
