In high dimensions, the classical Hotellingâ€™s *T*^{2} test tends to have low power or becomes undefined due to singularity of the sample covariance matrix. In this article, this problem is overcome by projecting the data matrix onto lower dimensional subspaces through multiplication by random matrices. We propose RAPTT (RAndom Projection T^{2}-Test), an exact test for equality of means of two normal populations based on projected lower dimensional data. RAPTT does not require any constraints on the dimension of the data or the sample size. A simulation study indicates that in high dimensions the power of this test is often greater than that of competing tests. The advantages of RAPTT are illustrated on a high-dimensional gene expression dataset involving the discrimination of tumor and normal colon tissues.