  Observational measures of the primordial helium mass fraction, $Y_{\rm P}$,
are of interest for cosmology and fundamental particle physics. Current
measures obtained from H II regions agree with the Standard Model prediction to
approximately 1% precision, although these determinations may be affected by
systematic uncertainties. This possibility can only be tested by independently
measuring the helium abundance in new ways. Here, we propose a novel method to
obtain a measurement of $Y_{\rm P}$ using hydrogen and helium recombination
line emission from RELHICs: pristine, gas-rich but star-free low-mass dark
matter haloes whose existence is predicted by hydrodynamical simulations.
Although expected to be uncommon and intrinsically faint in emission, the
primordial composition and simple physical properties of these objects make
them an ideal laboratory to determine $Y_{\rm P}$. We present radiative
transfer simulations to demonstrate the effectiveness of this approach, finding
that comparing the emission in H and He lines, either via their volumetric
emissivities, or integrated properties such as the surface brightness and total
flux, may be used to infer $Y_{\rm P}$. Furthermore, we show that RELHICs can
be used to provide an entirely novel constraint on the spectral slope of the
ultraviolet background, and discuss the possibility of measuring this slope and
the primordial helium abundance simultaneously.
