We present VRIJHK photometry, and optical and near-infrared spectroscopy, of the heavily extinguished Type Ia supernova (SN Ia) 2002cv, located in NGC 3190, which is also the parent galaxy of the SN Ia 2002bo. SN 2002cv, not visible in the blue, has a total visual extinction of 8.74 +/- 0.21 mag. In spite of this, we were able to obtain the light curves between -10 and +207 d from the maximum in the I band, and also to follow the spectral evolution, deriving its key parameters. We found the peak I-band brightness to be Imax = 16.57 +/- 0.10 mag, the maximum absolute I magnitude to be MmaxI = -18.79 +/- 0.20, and the parameter Δm15(B) specifying the width of the B-band light curve to be 1.46 +/- 0.17 mag. The latter was derived using the relations between this parameter and Δm40(I) and the time-interval Δtmax(I) between the two maxima in the I-band light curve. As has been found for previously observed, highly extinguished SNe Ia, a small value of 1.59 +/- 0.07 was obtained here for the ratio RV of the total-to-selective extinction ratio for SN 2002cv, which implies a small mean size for the grains along the line of sight towards us. Since it was found for SN 2002bo a canonical value of 3.1, here we present a clear evidence of different dust properties inside NGC 3190.