In a previous publication we have shown that the gauge theory of relativistic 3-Branes can be formulated in a conformally invariant way if the embedding space is six-dimensional. The implementation of conformal invariance requires the use of a modified measure, independent of the metric in the action. We here generalize the theory to include conformal invariance breaking and a dynamical scalar field with a non-trivial potential. The non conformal invariance contribution can be interpreted as originating from a continious "non ideal brane fluid" that exists between two singular branes. The scalar field potential also breaks the conformal invariance. At singular brane locations, conformal invariance is restored and the dynamics of the scalar field is frozen at a certain fixed value of the scalar field which depends on an arbitrary integration constant. Spontaneous Symmetry breaking can take place due to such boundary condition without the need of invoking tachyonic mass terms for the scalar field. In these Brane-world scenarios, zero 4-D cosmological constant is achieved without the need of invoking a fine tuned cosmological constant in 6D. Thus, no ``old'' cosmological constant problem appears. The use of a measure independent of the metric is crucial for obtaining all of the above results.