  Einstein's theory of general relativity states that clocks at different
gravitational potentials tick at different rates - an effect known as the
gravitational redshift. As fundamental probes of space and time, atomic clocks
have long served to test this prediction at distance scales from 30 centimeters
to thousands of kilometers. Ultimately, clocks will study the union of general
relativity and quantum mechanics once they become sensitive to the finite
wavefunction of quantum objects oscillating in curved spacetime. Towards this
regime, we measure a linear frequency gradient consistent with the
gravitational redshift within a single millimeter scale sample of ultracold
strontium. Our result is enabled by improving the fractional frequency
measurement uncertainty by more than a factor of 10, now reaching 7.6$\times
10^{-21}$. This heralds a new regime of clock operation necessitating
intra-sample corrections for gravitational perturbations.
