We study the presence of dark and bright modes in a planar metamaterial with a double rod unit cell introducing geometric asymmetry in rod lengths. The dark mode displays a Fano-type resonance with a sharp asymmetric profile, rendering it far more sensitive than the bright mode to slight variations of the dielectric environment. This peculiar feature may envisage the possible application of the asymmetric dimer metamaterial as an optical sensor for chemical or biological analysis, provided that the effect of material losses on the dark mode quality factor is properly taken into account.