Let p be a large prime number and g be any integer of multiplicative order T modulo p. We obtain a new estimate of the double exponential sumS=∑n∈N|∑m∈Mep(angm)|,gcd(a,p)=1, where N and M are intervals of consecutive integers with |N|=N and |M|=M≤T elements. One representative example is the following consequence of the main result: if N=M≈p1/3, then |S|<N2−1/8+o(1). We then apply our estimate to obtain new results on additive congruences involving intervals and exponential functions.