We consider the degenerate equation $$\partial_t f(t,x) - \partial_x \left( x^{\alpha} \partial_x f \right)(t,x) =0,$$ on the unit interval $x\in(0,1)$, in the strongly degenerate case $\alpha \in [1,2)$ with adapted boundary conditions at $x=0$ and boundary control at $x=1$. We use the flatness approach to construct explicit controls in some Gevrey classes steering the solution from any initial datum $f_0 \in L^2(0,1)$ to zero in any time $T>0$.