  NGC~613 is a nearby barred spiral galaxy with a nuclear ring. Exploiting high
spatial resolution ($\approx20$ pc) Atacama Large Millimeter/sub-millimeter
Array $^{12}$CO(1-0) observations, we study the giant molecular clouds (GMCs)
in the nuclear ring and its vicinity, identifying $158$ spatially- and
spectrally-resolved GMCs. The GMC sizes ($R_{\mathrm{c}}$) are comparable to
those of the clouds in the Milky Way (MW) disc, but their gas masses, observed
linewidths ($\sigma_{\mathrm{obs,los}}$) and gas mass surface densities are
larger. The GMC size -- linewidth relation ($\sigma_{\mathrm{obs,los}}\propto
R_{\mathrm{c}}^{0.77}$) is steeper than that of the clouds of the MW disc and
centre, and the GMCs are on average only marginally gravitationally bound (with
a mean virial parameter $\langle\alpha_{\mathrm{obs,vir}}\rangle\approx1.7$).
We discuss the possible origins of the steep size -- linewidth relation and
enhanced observed linewidths of the clouds and suggest that a combination of
mechanisms such as stellar feedback, gas accretion and cloud-cloud collisions,
as well as the gas inflows driven by the large-scale bar, may play a role.
