Abnormal β-amyloid (Aβ) accumulation in the brain is an early indicator of Alzheimer's disease and practical tests could help identify patients who could respond to treatment, now that promising anti-amyloid drugs are available. Even so, Aβ positivity (Aβ+) is assessed using PET or CSF assays, both highly invasive procedures. Here, we investigate how well Aβ+ can be predicted from T1 weighted brain MRI and gray matter, white matter and cerebrospinal fluid segmentations from T1-weighted brain MRI (T1w), a less invasive alternative. We used 3D convolutional neural networks to predict Aβ+ based on 3D brain MRI data, from 762 elderly subjects (mean age: 75.1 yrs. ± 7.6SD; 394F/368M; 459 healthy controls, 67 with MCI and 236 with dementia) scanned as part of the Alzheimer's Disease Neuroimaging Initiative. We also tested whether the accuracy increases when using transfer learning from the larger UK Biobank dataset. Overall, the 3D CNN predicted Aβ+ with 76% balanced accuracy from T1w scans. The closest performance to this was using white matter maps alone when the model was pre-trained on an age prediction in the UK Biobank. The performance of individual tissue maps was less than the T1w, but transfer learning helped increase the accuracy. Although tests on more diverse data are warranted, deep learned models from standard MRI show initial promise for Aβ+ estimation, before considering more invasive procedures.