(1) Background: Cardiac amyloidosis or “stiff heart syndrome” is a rare condition that occurs when amyloid deposits occupy the heart muscle. Many patients suffer from it and fail to receive a timely diagnosis mainly because the disease is a rare form of restrictive cardiomyopathy that is difficult to diagnose, often associated with a poor prognosis. This research analyses the characteristics of this pathology and proposes a statistical learning algorithm that helps to detect the disease. (2) Methods: The hospitalization clinical (medical and nursing ones) records used for this study are the basis of the learning and training techniques of the algorithm. The approach consisted of using the information generated by the patients in each admission and discharge episode and treating it as data vectors to facilitate their aggregation. The large volume of clinical histories implied a high dimensionality of the data, and the lack of diagnosis led to a severe class imbalance caused by the low prevalence of the disease. (3) Results: Although there are few patients with amyloidosis in this study, the proposed approach demonstrates that it is possible to learn from clinical records despite the lack of data. In the validation phase, the algorithm first acted on data from the general study population. It then was applied to a sample of patients diagnosed with heart failure. The results revealed that the algorithm detects disease when data vectors profile each disease episode. (4) Conclusions: The prediction levels showed that this technique could be useful in screening processes on a specific population to detect the disease.