  We prove the birational rigidity of Fano complete intersections of index 1
with a singular point of high multiplicity, which can be close to the degree of
the variety. In particular, the groups of birational and biregular
automorphisms of these varieties are equal, and they are non-rational. The
proof is based on the techniques of the method of maximal singularities, the
generalized $4n^2$-inequality for complete intersection singularities and the
technique of hypertangent divisors.
