The magnetic phase diagram of a spin-1 2 chain antiferromagnet Sr 2 CuO 3 is studied by an ultrasound phasesensitive detection technique. The system is in the extreme proximity of the Luttinger-liquid quantum-critical point and we observe an unusually strong effect of magnetic field, which is very weak compared to the in-chain interaction, on the Néel ordering temperature. Inside the ordered phase, we detect an unexpected, field-induced continuous phase transition. The transition is accompanied by softening of magnetic excitation observed by electron-spin resonance, which in previous work [E. G. Sergeicheva et al., Phys. Rev. B 95, 020411(R) ( 2017)] was associated with a longitudinal (amplitude) mode of the order parameter. These results suggest a transition from a transverse collinear antiferromagnet to an amplitude-modulated spin-density-wave phase in a very weak magnetic field, which is unexpected for a system of weakly coupled Heisenberg spin-1 2 chains.