  We compute the Newton--Okounkov bodies of line bundles on the complete flag
variety of GL_n for a geometric valuation coming from a flag of translated
Schubert subvarieties. The Schubert subvarieties correspond to the terminal
subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of
the longest element in the Weyl group. The resulting Newton--Okounkov bodies
coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.
