We investigate an asymmetric zigzag spin ladder with different exchange int
egrals on both legs using bosonization and renormalization group approaches
. When the leg exchange integrals and frustration both are sufficiently sma
ll, renormalization group analysis shows that the Heisenberg critical point
flows to an intermediate-coupling fixed point with gapless excitations and
a vanishing spin velocity. When they are large, a spin gap opens and a dim
er liquid is realized. Here, we find a continuous manifold of Hamiltonians
with dimer product ground states, interpolating between the Majumdar-Ghosh
and sawtooth spin-chain model.