  Optical atomic clocks$^{1,2}$ use electronic energy levels to precisely keep
track of time. A clock based on nuclear energy levels promises a
next-generation platform for precision metrology and fundamental physics
studies. Thorium-229 nuclei exhibit a uniquely low energy nuclear transition
within reach of state-of-the-art vacuum ultraviolet (VUV) laser light sources
and have therefore been proposed for construction of the first nuclear
clock$^{3,4}$. However, quantum state-resolved spectroscopy of the $^{229m}$Th
isomer to determine the underlying nuclear structure and establish a direct
frequency connection with existing atomic clocks has yet to be performed. Here,
we use a VUV frequency comb to directly excite the narrow $^{229}$Th nuclear
clock transition in a solid-state CaF$_2$ host material and determine the
absolute transition frequency. We stabilize the fundamental frequency comb to
the JILA $^{87}$Sr clock$^2$ and coherently upconvert the fundamental to its
7th harmonic in the VUV range using a femtosecond enhancement cavity. This VUV
comb establishes a frequency link between nuclear and electronic energy levels
and allows us to directly measure the frequency ratio of the $^{229}$Th nuclear
clock transition and the $^{87}$Sr atomic clock. We also precisely measure the
nuclear quadrupole splittings and extract intrinsic properties of the isomer.
These results mark the start of nuclear-based solid-state optical clock and
demonstrate the first comparison of nuclear and atomic clocks for fundamental
physics studies. This work represents a confluence of precision metrology,
ultrafast strong field physics, nuclear physics, and fundamental physics.
