  The shape of the electron spectrum emitted in $\beta$ decay carries a wealth
of information about nuclear structure and fundamental physics. In spite of
that, few dedicated measurements have been made of $\beta$-spectrum shapes. In
this work we present a newly developed detector for $\beta$ electrons based on
a telescope concept. A thick plastic scintillator is employed in coincidence
with a thin silicon detector. The first measurements employing this detector
have been carried out with mono-energetic electrons from the high-energy
resolution electron-beam spectrometer at Bordeaux. Here we report on the good
reproduction of the experimental spectra of mono-energetic electrons using
Monte Carlo simulations. This is a crucial step for future experiments, where a
detailed Monte Carlo characterization of the detector is needed to determine
the shape of the $\beta$-electron spectra by deconvolution of the measured
spectra with the response function of the detector. A chamber to contain two
telescope assemblies has been designed for future $\beta$-decay experiments at
the Ion Guide Isotope Separator On-Line facility in Jyv\"askyl\"a, aimed at
improving our understanding of reactor antineutrino spectra.
