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 $A_{V} = 0.14pm 0.01$ for a total dust mass, $M_{d} = 2.5pm0.2times10^{9}M_{odot}$, hence a dust-to-gas mass ratio $M_{d}/M_{g} = 3.0pm 0.3 times 10^{-4}$. Based on the upper limits on the flux density $mathrm{I_{250mu 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.
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