  A nonperturbative analytical description of the SU(2)_1 quantum critical
point in an explicitly dimerized two-leg spin-1/2 Heisenberg ladder is
presented. It is shown that this criticality essentially coincides with that
emerging in a weakly dimerized spin-1 chain with a small Haldane gap. The
approach is based on the mapping onto an SO(3)-symmetric model of three
strongly coupled quantum Ising chains. This mapping is used to establish the
correspondence between all physical fields of the spin ladder and those
characterizing the SU(2)_1 criticality at the infrared fixed point.
