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Summary form only given: Measurements of magnetic fields are a key issue in numerous studies of plasmas carrying high-current pulses. However, the traditional techniques, based on Zeeman splitting or Faraday rotation, can not be used in situations where the magnetic field has no definite direction during the time of observation, and/or at the region viewed by the diagnostic system, as commonly occurs in laser-produced or pinch plasmas. This problem is also manifested for magnetic fields with amplitudes that vary in time or space. We present a new spectroscopic diagnostics that allows for determining the magnetic field straight even in situations where the field direction and amplitude vary in space and during the time of observation. Furthermore, it is demonstrated that this technique allows for accurate B-field detection in the presence of other broadening mechanisms in the plasma (Stark and Doppler effects). The technique is based on the spectroscopic analysis of line-shapes of different fine-structure components of the same atomic multiplet that undergo different splitting under the B-field. The approach allows for distinguishing between different mechanisms influencing the line-shapes. Here we report on the experimental verification of the proposed method for the B-field measurements using laser-produced plasma injected into the A-K gap of a vacuum transmission line carrying a 160 kA, 100 ns current. The magnetic field generated in the inter-electrode gap produces line splitting of the plasma emission lines, that is unambiguously inferred from the emission line shape in spite of the dominance of the line shape by the Stark broadening.