  We present the first detection of diffuse dust in the intra-cluster medium of
the Virgo cluster out to $\sim$0.4 virial radii, and study the radial variation
of its properties on a radial scale of the virial radius. Analysing near-UV -
$i$ colours for a sample of $\sim12000$ background galaxies with redshifts
$0.02 < z < 0.8$, we find significant colour reddening and relate it to
variation in $E(B-V)$ values. The $E(B-V)$ mean profile shows a dust component
characterised by an average reddening $E(B-V)\sim0.042 \pm 0.004$ mag within
1.5 degrees ($\sim0.3\, r_{vir}$) from the cluster centre. Assuming a Large
Magellanic Cloud extinction law, we derive an average visual extinction $A_{V}
= 0.14\pm 0.01$ for a total dust mass, $M_{d} =
2.5\pm0.2\times10^{9}M_{\odot}$, hence a dust-to-gas mass ratio $M_{d}/M_{g} =
3.0\pm 0.3 \times 10^{-4}$. Based on the upper limits on the flux density
$\mathrm{I_{250\mu m} = 0.1\, MJy sr^{-1}} $ derived from $Herschel$ data, we
estimate an upper limit for the dust temperature of $T_{d} \sim 10\, K$.
However, similar densities can be obtained with dust at higher temperatures
with lower emissivities. The Virgo cluster has diffuse dust in its
intra-cluster medium characterised by different physical properties as those
characterising the Milky Way dust. The diffuse dust in Virgo is transported
into the cluster space through similar phenomena (stripping) as those building
up the optical intra-cluster light, and it constitutes an additional cooling
agent of the cluster gas.
