  In this manuscript, we study the limiting distribution for the joint law of
the largest and the smallest singular values for random circulant matrices with
generating sequence given by independent and identically distributed random
elements satisfying the so-called Lyapunov condition. Under an appropriated
normalization, the joint law of the extremal singular values converges in
distribution, as the matrix dimension tends to infinity, to an independent
product of Rayleigh and Gumbel laws. The latter implies that a normalized
condition number converges in distribution to a Fr\'echet law as the dimension
of the matrix increases.
