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In this paper we build a Continuous Wavelet Transform (CWT) on the upper sheet of the 2-hyperboloid . First, we define a class of suitable dilations on the hyperboloid through conic projection. Then, incorporating hyperbolic motions belonging to , we define a family of hyperbolic wavelets. The continuous wavelet transform is obtained by convolution of the scaled wavelets with the signal. The wavelet transform is proved to be invertible whenever wavelets satisfy a particular admissibility condition, which turns out to be a zero-mean condition. We then provide some basic examples and discuss the limit at null curvature.