  We investigate the ground-state properties of the spin-half $J_1{-}J_2$
Heisenberg chain with a next-nearest-neighbor spin-Peierls dimerization using
conformal field theory and Lanczos exact diagonalizations. In agreement with
the results of a recent bosonization analysis by Sarkar and Sen, we find that
for small frustration ($J_2/J_1$) the system is in a Luttinger spin-fluid
phase, with gapless excitations, and a finite spin-wave velocity. In the regime
of strong frustration the ground state is spontaneously dimerized and the bond
alternation reduces the triplet gap, leading to a slight enhancement of the
critical point separating the Luttinger phase from the gapped one. An accurate
determination of the phase boundary is obtained numerically from the study of
the excitation spectrum.
