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Dual-comb spectroscopy is a rapidly developing technique enabling ultraprecise broadband optical diagnostics of atoms and molecules. This powerful tool typically requires two phase-locked femtosecond lasers, yet it has been shown that it can be realized without any stabilization if the combs are generated from a single laser cavity. Still, unavoidable intrinsic relative phase-fluctuations always set a limit on the precision of any spectroscopic measurements, hitherto limiting the applicability of bulk dual-comb lasers for mode-resolved studies. Here, we demonstrate a versatile concept for low-noise dual-comb generation from a single-cavity femtosecond solid-state laser based on intrinsic polarization-multiplexing inside an optically anisotropic gain crystal. Due to intracavity spatial separation of the orthogonally-polarized beams, two sub-100 fs pulse trains are simultaneously generated from a 1.05 $μ$m Yb:CNGS oscillator with a repetition rate difference of 4.7 kHz. The laser exhibits the lowest relative noise ever demonstrated for a bulk dual-comb source, supporting free-running mode-resolved spectroscopic measurements over a second. Moreover, the developed dual-comb generation technique can be applied to any solid-state laser exploiting a birefringent active crystal, paving the way towards a new class of highly-coherent, single-cavity, dual-comb laser sources operating in various spectral regions.