  Direct production of the charmonium-like state $X(3872)$ in $e^+e^-$
collisions is considered in the framework of the vector meson dominance model.
An order-of-magnitude estimate for the width $\Gamma(X\to e^+e^-)$ is found to
be $\gtrsim$0.03 eV. The same approach applied to the $\chi_{c1}$ charmonium
decay predicts the corresponding width of the order 0.1 eV in agreement with
earlier estimates. Experimental perspectives for the direct production of the
$1^{++}$ charmonia in $e^+e^-$ collisions are briefly discussed.
