After a stroke, approximately one-third of patients suffer from aphasia, a language disorder that impairs communication ability. The standard behavioral tests used to diagnose aphasia are time-consuming, require subjective interpretation, and have low ecological validity. As a consequence, comorbid cognitive problems present in individuals with aphasia can bias test results, generating a discrepancy between test outcomes and everyday-life language abilities. Neural tracking of the speech envelope is a promising tool for investigating brain responses to natural speech. The envelope of speech is crucial for speech understanding, encompassing cues for detecting and segmenting linguistic units, e.g., phrases, words and phonemes. In this study, we aimed to test the potential of the neural envelope tracking technique for detecting language impairments in aphasia. We recorded EEG from 27 individuals with aphasia in the chronic phase after stroke and 22 healthy controls while they listened to a 25-minute story. We quantified neural envelope tracking in a broadband frequency range as well as in the delta, theta, alpha, beta, and gamma frequency bands using mutual information analysis. Besides group differences in neural tracking measures, we also tested its suitability for detecting aphasia at the individual level using a support vector machine classifier. We further investigated the required recording length for the classifier to detect aphasia and to obtain reliable outcomes. Individuals with aphasia displayed decreased neural envelope tracking compared to healthy controls in the broad, delta, theta, and gamma band, which is in line with the assumed role of these bands in auditory and linguistic processing of speech. Neural tracking in these frequency bands effectively captured aphasia at the individual level, with a classification accuracy of 84% and an area under the curve of 88%. Moreover, we demonstrated that high-accuracy detection of aphasia can be achieved in a time-efficient (5-7 minutes) and highly reliable manner (split-half reliability correlations between R=0.62 and R=0.96 across frequency bands). Our study shows that neural envelope tracking of natural speech is an effective biomarker for language impairments in post-stroke aphasia. We demonstrated its potential as a diagnostic tool with high reliability, individual-level detection of aphasia, and time-efficient assessment. This work represents a significant step towards more automatic, objective, and ecologically valid assessments of language impairments in aphasia.