The birth of the first luminous sources and the ensuing epoch of reionization are best studied via the redshifted 21-cm emission line, the signature of the first two imprinting the last. In this work, we present a fully Bayesian method, hibayes, for extracting the faint, global (sky-averaged) 21-cm signal from the much brighter foreground emission. We show that a simplified (but plausible) Gaussian model of the 21-cm emission from the Cosmic Dawn epoch (15 ≲ z ≲ 30), parametrized by an amplitude $A_{\rm H\,\small {I}}$, a frequency peak $\nu _{\rm H\,\small {I}}$ and a width $\sigma _{\rm H\,\small {I}}$, can be extracted even in the presence of a structured foreground frequency spectrum (parametrized as a seventh-order polynomial), provided sufficient signal-to-noise (400 h of observation with a single dipole). We apply our method to an early, 19-min-long observation from the Large aperture Experiment to detect the Dark Ages, constraining the 21-cm signal amplitude and width to be $-890 \lt A_{\rm H\,\small {I}} \lt 0$ mK and $\sigma _{\rm H\,\small {I}} \gt 6.5$ MHz (corresponding to Δz > 1.9 at redshift z ≃ 20) respectively at the 95-per cent confidence level in the range 13.2 < z < 27.4 (100 > ν > 50 MHz).