  We establish the weak convergence of the intensity of a nearly-unstable
Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling
limit for a financial market model where orders to buy or sell an asset arrive
according to a Hawkes process with power-law kernel. After suitable rescaling
the price-volatility process converges weakly to a rough Heston model. Our
convergence result is much stronger than previously established ones that have
either focused on light-tailed kernels or the convergence of integrated
volatility process. The key is to establish the tightness of the family of
rescaled volatility processes. This is achieved by introducing a new methods to
establish the $C$-tightness of c\`adl\`ag processes based on the classical
Kolmogorov-Chentsov tightness criterion for continuous processes.
