[en] It is done a ''minimal'' change in the existing 4-dimensional relativity, by extending it to the 6-dimensional conformal (etasup(a))-space (flat or curved one) with the metric tensor gsub(ab) (a, b=0, 1, 2, 3, 5, 6), where all components of the 6-vector etasup(a)=(etasup(μ)=kxsup(μ), k, lambda) are considered as independent physical degrees of freedom. All basic equations of (special and general) relativity in 6-dimensional (flat or curved) conformal (etasup(a))-space have the same form as the corresponding equations in the 4-dimensional space. The novel feature of such an extended theory (named ''conformal relativity'') is the introduction of the scale degree of freedom k, which can be different from 1 and can change along the particle world-line. However, if k=1, then the conformal relativity reduces to the usual 4-dimensional relativity. Geodesics in the curved (etasup(a))-space correspond to the motion of electrically charged test particles in gravitational and/or electromagnetic fields. The field equations for the metric tensor gsub(ab) are Einstein equations, extended to the (etasup(a))-space; they describe a gravitational and electromagnetic field