  Third-order weak lensing statistics are a promising tool for cosmological
analyses since they extract cosmological information in the non-Gaussianity of
the cosmic large-scale structure. However, such analyses require precise and
accurate models for the covariance. In this second paper of a series on
third-order weak lensing statistics, we derive and validate an analytic model
for the covariance of the third-order aperture statistics $\langle
M_\mathrm{ap}^3\rangle$. We derive the covariance model from a real-space
estimator for $\langle M_\mathrm{ap}^3\rangle$. We validate the model by
comparing it to estimates from simulated Gaussian random fields (GRF) and two
sets of N-body simulations. Finally, we perform mock cosmological analyses with
the model covariance and the simulation estimate to compare the resulting
parameter constraints. We find good agreement between the model and the
simulations, both for the GRF and the $N$-body simulations. The figure-of-merit
in the $S_8$-$\Omega_\mathrm{m}$ plane from our covariance model is within 3\%
of the one obtained from the simulated covariances. We also show that our
model, which is based on an estimator using convergence maps, can be used to
obtain upper and lower bounds for the covariance of an estimator based on
three-point shear correlation functions. This second estimator is required for
realistic survey data. In our derivation, we find that the $\langle
M_\mathrm{ap}^3\rangle$ covariance cannot be obtained from the bispectrum
covariance and that it includes several `finite-field terms' that do not scale
with the inverse survey area. Our covariance model is sufficiently accurate for
analysing stage III surveys. Covariances for statistics in Fourier space cannot
always be straightforwardly converted into covariance for real-space
statistics. The modelling code is available at
https://github.com/sheydenreich/threepoint/releases/ .
