We seek to evaluate the detection performance of a rapid primary screening tool of Covid-19 solely based on the cough sound from 8,380 clinically validated samples with laboratory molecular-test (2,339 Covid-19 positives and 6,041 Covid-19 negatives). Samples were clinically labeled according to the results and severity based on quantitative RT-PCR (qRT-PCR) analysis, cycle threshold, and lymphocytes count from the patients. Our proposed generic method is an algorithm based on Empirical Mode Decomposition (EMD) with subsequent classification based on a tensor of audio features and a deep artificial neural network classifier with convolutional layers called DeepCough'. Two different versions of DeepCough based on the number of tensor dimensions, i.e. DeepCough2D and DeepCough3D, have been investigated. These methods have been deployed in a multi-platform proof-of-concept Web App CoughDetect to administer this test anonymously. Covid-19 recognition results rates achieved a promising AUC (Area Under Curve) of 98.800.83%, sensitivity of 96.431.85%, and specificity of 96.201.74%, and 81.08%5.05% AUC for the recognition of three severity levels. Our proposed web tool and underpinning algorithm for the robust, fast, point-of-need identification of Covid-19 facilitates the rapid detection of the infection. We believe that it has the potential to significantly hamper the Covid-19 pandemic across the world.