  At the Karlsruhe pulsed 3.75\,MV Van de Graaff accelerator the thermonuclear
$^{48}$Ca(n,$\gamma$)$^{49}$Ca(8.72\,min) cross section was measured by the
fast cyclic activation technique via the 3084.5\,keV $\gamma$-ray line of the
$^{49}$Ca-decay. Samples of CaCO$_3$ enriched in $^{48}$Ca by 77.87\,\% were
irradiated between two gold foils which served as capture standards. The
capture cross-section was measured at the neutron energies 25, 151, 176, and
218\,keV, respectively. Additionally, the thermal capture cross-section was
measured at the reactor BR1 in Mol, Belgium, via the prompt and decay
$\gamma$-ray lines using the same target material. The
$^{48}$Ca(n,$\gamma$)$^{49}$Ca cross-section in the thermonuclear and thermal
energy range has been calculated using the direct-capture model combined with
folding potentials. The potential strengths are adjusted to the scattering
length and the binding energies of the final states in $^{49}$Ca. The small
coherent elastic cross section of $^{48}$Ca+n is explained through the nuclear
Ramsauer effect. Spectroscopic factors of $^{49}$Ca have been extracted from
the thermal capture cross-section with better accuracy than from a recent (d,p)
experiment. Within the uncertainties both results are in agreement. The
non-resonant thermal and thermonuclear experimental data for this reaction can
be reproduced using the direct-capture model. A possible interference with a
resonant contribution is discussed. The neutron spectroscopic factors of
$^{49}$Ca determined from shell-model calculations are compared with the values
extracted from the experimental cross sections for $^{48}$Ca(d,p)$^{49}$Ca and
$^{48}$Ca(n,$\gamma$)$^{49}$Ca.
