Environmental fluctuations constitute a major source of decoherence in quantum devices, rendering their spectral characterization essential for robust control and error correction. Existing qubit noise spectroscopy methods often rely on elaborate control sequences or extensive scans over evolution time. Here we introduce a noise spectroscopy protocol based on repetitive weak measurements, implemented via Ramsey interferometry on a probe qubit and interleaved with free environmental evolution. A quantum channel analysis shows that, in the weak-measurement regime, correlations between the first and subsequent qubit measurement outcomes approximately sample the symmetrized environmental noise correlation function, up to an overall scale factor and a weak damping envelope induced by measurement backaction. A Fourier transform of these correlations reconstructs the noise spectrum across the sampled bandwidth, without assuming Gaussian fluctuations or requiring probe coherence throughout the full sampling interval. For $N$ uniformly spaced sampling times, the accumulated evolution time scales linearly with $N$, rather than quadratically as in conventional correlation spectroscopy. Simulations of an Ohmic bosonic bath and an interacting nuclear spin cluster recover their characteristic spectral features. These results demonstrate the potential of sequential weak measurements to enable resource-efficient broadband spectroscopy of long-lived quantum dephasing environments under suitable parameter regimes, albeit subject to measurement backaction and other possible experimental limitations.