A single annular ring resonator is proposed to design a dual-mode dual-band bandpass filter (BPF) with adjustable frequency ratio of the two passbands. The two passbands are formed by the first two pairs of degenerate modes of an annular ring resonator. The change of the inner concentric circular radius leads to the variation of surface current distributions of the and resonant modes accordingly. As a result, the frequency ratio between 1.65 and 2.00 can be achieved when it changes between a circular patch and a ring resonator. The resonator is inductively coupled to the input and output ports by a pair of high impedance lines. To verify the design principle, two filters with different frequency ratios of the two passbands, 1.75 and 1.90, are designed, fabricated and measured. The experimental results agree well with the predicted ones.