The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the $ν$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(ν)\geq 1$ is the largest natural number satisfying $\int_0^1 α^{2j}ν(dα)=(2j+1)^{-1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.