The Hermes experiment has investigated the tensor spin structure of the deuteron using the 27.6 GeV/c positron beam of DESY HERA. The use of a tensor-polarized deuteron gas target with only a negligible residual vector polarization enabled the first measurement of the tensor asymmetry A(d)zz and the tensor structure function b(d)1 for average values of the Bjorken variable 0.01< <x> <0.45 and of the negative of the squared four-momentum transfer 0.5 GeV2 < <Q2> <5 GeV2. The quantities A(d)zz and b(d)1 are found to be nonzero. The rise of b(d)1 for decreasing values of x can be interpreted to originate from the same mechanism that leads to nuclear shadowing in unpolarized scattering.