  Coherence, the stability of the relative phase between quantum states, lies
at the heart of quantum mechanics. Applications such as precision measurement,
interferometry, and quantum computation are enabled by physical systems that
have quantum states with robust coherence. With the creation of molecular
ensembles at sub-$\mu$K temperatures, diatomic molecules have become a novel
system under full quantum control. Here, we report on the observation of stable
coherence between a pair of nuclear spin states of ultracold fermionic NaK
molecules in the singlet rovibrational ground state. Employing microwave
fields, we perform Ramsey spectroscopy and observe coherence times on the scale
of one second. This work opens the door for the exploration of single molecules
as a versatile quantum memory. Switchable long-range interactions between
dipolar molecules can further enable two-qubit gates, allowing quantum storage
and processing in the same physical system. Within the observed coherence time,
$10^4$ one- and two-qubit gate operations will be feasible.
