We have used high-resolution magnetic birefringence to study very dilute mixtures of the nonionic surfactant pentaethylene glycol mono-dodecyl-ether (C12E5) in water at fields to H=18 T. The induced refractive index anisotropy Delta n shows distinct H dependencies in the L(alpha)-lamellar and L(3)-"sponge" phases, and demonstrates that these are separated by a two-phase coexistence region. In the L(3) phase and for surfactant concentrations gamma similar or equal to 0.02 (the vicinity of the L(alpha) phase terminus), we observe a sharp change in the magnitude and H dependence of Delta n. A quantitative analysis shows this to be consistent with a crossover from a highly-connected "symmetric sponge" to a droplet or "asymmetric sponge" structure as a function of gamma, a result previously suggested by neutron scattering and electrical conductivity measurements [R. Strey, R. Schomacker, D. Roux, F. Nallet, and U. Olssen, J. Chem. Soc. Faraday Trans. II 86, 2253 (1990)]. Finally, for gamma greater than or similar to 0.02 and at large H, we present evidence of a field-induced transformation of the L(3) structure.