Based on wavelet multi-resolution features and signal multi-channel theory, results on minimum-energy wavelet frames are presented for intervals with arbitrary integer dilation factors not less than 2. First, we define the concept of minimum-energy wavelet frames on such intervals, give the left and right boundary scaling functions and the left and right boundary wavelet frame functions, and present the necessary and sufficient conditions for the minimum-energy wavelet frame on the interval. Second, the design of the construction algorithm of minimum-energy wavelet frame on the interval with arbitrary integer support is discussed, and the structure of the matrices used in the algorithm is presented constructively. Third, the decomposition and reconstruction formulas of the minimum-energy wavelet frame on the interval are deduced. These are similar to those for orthogonal wavelets on the interval. Finally, based on B-spline functions, several numerical examples are given.