  We propose a two-qubit gate based on dipolar exchange interactions between
individually addressable ultracold polar molecules in an array of optical
dipole traps. Our proposal treats the full Hamiltonian of the $^1\Sigma^+$
molecule NaCs, utilizing a pair of nuclear spin states as storage qubits. A
third rotationally excited state with rotation-hyperfine coupling enables
switchable dipolar exchange interactions between two molecules to generate an
iSWAP gate. All three states are insensitive to external magnetic and electric
fields. Impacts on gate fidelity due to coupling to other molecular states,
imperfect ground-state cooling, blackbody radiation and vacuum spontaneous
emission are small, leading to potential fidelity above $99.99~\%$ in a
coherent quantum system that can be scaled by purely optical means.
