  We report the structural and magnetic properties of high-quality bulk single
crystals of the kagome ferromagnet Fe$_3$Sn. The dependence of magnetisation on
the magnitude and orientation of the external field reveals strong easy-plane
type uniaxial magnetic anisotropy, which shows a monotonous increase from
$K_1=-0.99\times 10^6 J/m^3$ at 300\,K to $-1.23\times10^6 J/m^3$ at 2\,K. Our
\textit{ab initio} electronic structure calculations yield the value of total
magnetic moment of about 6.9 $\mu_B$/f.u. and a magnetocrystalline anisotropy
energy density of 0.406\,meV/f.u. ($1.16\times10^6 J/m^3$) both being in good
agreement with the experimental values. The self-consistent DFT computations
for the components of the spin/orbital moments indicate that the small
difference between the saturation magnetisations measured along and
perpendicular to the kagome layers results from the subtle balance between the
Fe and Sn spin/orbital moments on the different sites. In zero field, magnetic
force microscopy reveals micrometer-scale magnetic vortices with weakly pinned
cores that vanish at $\sim$3\,T applied perpendicular to the kagome plane. Our
micromagnetic simulations, using the experimentally determined value of
anisotropy, well reproduce the observed vortex-domain structure. The present
study, in comparison with the easy-axis ferromagnet Fe$_3$Sn$_2$, shows that
varying the stacking of kagome layers provides an efficient control over
magnetic anisotropy in this family of Fe-based kagome magnets.
