The problem of weak signal detection in Gaussian noise is addressed in the Neyman-Pearson framework with compressive measurements. A locally optimum detector is first devised assuming that the signal is nonsparse by approximating the test statistic around zero using a Taylor series, which is a good estimate only in a small radius around zero. When the signal is sparse, it is shown that the performance of this test degrades. To improve its performance, a new test is devised by deriving the Padé approximation of the test statistic around zero. Padé approximants estimate functions as the rational quotient of two lower degree polynomials and consistently have a wider radius of convergence than the Taylor series. The performance of the Padé-approximated test is better than its Taylor series counterpart and is comparable to the conventional locally optimum test with uncompressed measurements. Simulation results are presented to support the analytical findings of the work.