  We report on a search for electromagnetic and/or hadronic showers (cascades)
induced by a diffuse flux of neutrinos with energies between 5 TeV and 300 TeV
from extraterrestrial sources. Cascades may be produced by matter interactions
of all flavors of neutrinos, and contained cascades have better energy
resolution and afford better background rejection than through-going
\nu_\mu-induced muons. Data taken in 1997 with the AMANDA detector were
searched for events with a high-energy cascade-like signature. The observed
events are consistent with expected backgrounds from atmospheric neutrinos and
catastrophic energy losses from atmospheric muons. Effective volumes for all
flavors of neutrinos, which allow the calculation of limits for any neutrino
flux model, are presented. The limit on cascades from a diffuse flux of
\nu_e+\nu_\mu+\nu_\tau+\bar{\nu}_e+\bar{\nu}_\mu+\bar{\nu}_\tau is E^2 d\Phi/dE
< 9.8 x 10^{-6} GeV/(cm^2 s sr), assuming a neutrino flavor flux ratio of 1:1:1
at the detector. The limit on cascades from a diffuse flux of \nu_e+\bar{\nu}_e
is E^ d\Phi/dE < 6.5 x 10{-6} GeV/(cm^2 s sr), independent of the assumed
neutrino flavor flux ratio.
