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This paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments (MoM). The basis is derived from orthogonal Legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features. Expressions are presented for wire, surface, and volume elements but emphasis is given to the surface elements. In this case, the new hierarchical basis leads to a near-orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge. In addition, all higher order terms in the expansion have two vanishing moments. In contrast to existing formulations, these properties allow the use of very high-order basis functions without introducing ill-conditioning of the resulting MoM matrix. Numerical results confirm that the condition number of the MoM matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions. As a consequence of the excellent condition numbers, we demonstrate that even very high-order MoM systems, e.g., tenth order, can be solved efficiently with an iterative solver in relatively few iterations.