  We compute R\'enyi entropies for a spherical entangling surface in
four-dimensional N=4 super-Yang-Mills at strong coupling using the AdS/CFT
correspondence. Incorporating the effects of the leading \alpha' corrections to
the low energy effective action of type IIB string theory, we calculate the
leading corrections in inverse powers of the 't Hooft coupling (and the number
of colours). The results are compared with known weak coupling calculations.
Setting the order of the R\'enyi entropy q to one, it reduces to the
entanglement entropy and the strong and weak coupling results match without any
corrections, as expected. In the limit of q going to 0, the relation between
the strong and weak coupling entropies is connected to the known corrections
for the thermal free energy in flat space. We also compute the correction to
the scaling dimension of twist operators.
