Galaxies with bright rings, little gas, and little or no obvious bar are not adequately explained by current theories of ring formation in disk galaxies. This series focuses on four extreme examples which illustrate the diverging characteristics well. The galaxies are studied by means of multicolor CCD surface photometry obtained with the 3.9 m telescope of the Anglo-Australian Observatory. The ringed Seyfert galaxy NGC 3081 is an SO/a system possessing an unusually high contrast inner ring enveloping a weak bar. In this first paper of the series, the structure of this galaxy is described using BVRCIC surface photometry. The observations reveal a complex system which shares many properties in common with NGC 1433, a prototype SB(r) type spiral with a strong bar. The bar of NGC 3081 is much weaker than that in NGC 1433, but the inner ring is more eccentric than those of most SB galaxies. Significant azimuthal variations in the color indices occur around the ring which are probably due to the elongated shape. It is suggested that while many properties of the three rings in this galaxy can be explained in terms of orbit resonance theory, the extreme shape of the inner ring and the weakness of the bar are contradictions to the theory that need to be addressed. It is suggested that NGC 3081 is a possible example of an object where secular dissolution of the bar is occurring.