  Let $B=(B^{(1)},B^{(2)})$ be a two-dimensional fractional Brownian motion
with Hurst index $\alpha\in (0,1/4)$. Using an analytic approximation $B(\eta)$
of $B$ introduced in \cite{Unt08}, we prove that the rescaled L\'evy area
process $(s,t)\to \eta^{\half(1-4\alpha)}\int_s^t dB_{t_1}^{(1)}(\eta)
\int_s^{t_1} dB_{t_2}^{(2)}(\eta)$ converges in law to $W_t-W_s$ where $W$ is a
Brownian motion independent from $B$. The method relies on a very general
scheme of analysis of singularities of analytic functions, applied to the
moments of finite-dimensional distributions of the L\'evy area.
