  A new strong-interaction has been postulated for neutrinos above ~10^{19} eV
to explain the production of highest-energy cosmic ray events. We derive a
dispersion relation relating the hypothesized high-energy cross-section to the
lower-energy neutrino-nucleon elastic amplitude. Remarkably, we find that the
real forward amplitude becomes anomalous seven orders of magnitude lower in
energy than does the total cross-section. We discuss possible measurable
consequences of this early onset of new neutrino physics, and conclude that a
significantly enhanced elastic \nu-N scattering rate may occur for the neutrino
beams available at Fermilab and CERN.
