Aims. We present the first detection of diffuse dust in the intra-cluster medium of the Virgo cluster out to ∼0.4 virial radii, and study the radial variation of its properties on a radial scale of the virial radius. Methods. Analysing near-UV – i colours for a sample of ∼12 000 background galaxies with redshifts 0.02 < z < 0.8, we find significant colour reddening and relate it to variation in E(B − V) values. Results. The E(B − V) mean profile shows a dust component characterised by an average reddening E(B − V) ∼ 0.042 ± 0.004 mag within 1.5 degrees (∼0.3 r_vir) from the cluster centre. Assuming a Large Magellanic Cloud extinction law, we derive an average visual extinction A_V = 0.14 ± 0.01 for a total dust mass, M_d = 2.5 ± 0.2 × 10⁹ M_⊙, hence a dust-to-gas mass ratio M_d/M_g = 3.0 ± 0.3 × 10⁻⁴. Based on the upper limits on the flux density I_{250μm = 0.1 MJy sr⁻¹} derived from Herschel data, we estimate an upper limit for the dust temperature of T_d ∼ 10 K. However, similar densities can be obtained with dust at higher temperatures with lower emissivities. Conclusions. 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.