  In a recent paper A. Fraser and R. Schoen have proved the existence of free
boundary minimal surfaces $\Sigma\_n$ in $B^3$ which have genus $0$ and $n$
boundary components, for all $ n \geq 3$. For large $n$, we give an independent
construction of $\Sigma\_n$ and prove the existence of free boundary minimal
surfaces $\tilde \Sigma\_n$ in $B^3$ which have genus $1$ and $n$ boundary
components. As $n$ tends to infinity, the sequence $\Sigma\_n$ converges to a
double copy of the unit horizontal (open) disk, uniformly on compacts of $B^3$
while the sequence $\tilde \Sigma\_n$ converges to a double copy of the unit
horizontal (open) punctured disk, uniformly on compacts of $B^3-\{0\}$.
