Ultra-high resolution 7 tesla (7T) magnetic resonance imaging (MRI) provides detailed anatomical views, offering better signal-to-noise ratio, resolution and tissue contrast than 3T MRI, though at the cost of accessibility. We present an advanced deep learning model for synthesizing 7T brain MRI from 3T brain MRI. Paired 7T and 3T T1-weighted images were acquired from 172 participants (124 cognitively unimpaired, 48 impaired) from the Swedish BioFINDER-2 study. To synthesize 7T MRI from 3T images, we trained two models: a specialized U-Net, and a U-Net integrated with a generative adversarial network (GAN U-Net). Our models outperformed two additional state-of-the-art 3T-to-7T models in image-based evaluation metrics. Four blinded MRI professionals judged our synthetic 7T images as comparable in detail to real 7T images, and superior in subjective visual quality to 7T images, apparently due to the reduction of artifacts. Importantly, automated segmentations of the amygdalae of synthetic GAN U-Net 7T images were more similar to manually segmented amygdalae (n=20), than automated segmentations from the 3T images that were used to synthesize the 7T images. Finally, synthetic 7T images showed similar performance to real 3T images in downstream prediction of cognitive status using MRI derivatives (n=3,168). In all, we show that synthetic T1-weighted brain images approaching 7T quality can be generated from 3T images, which may improve image quality and segmentation, without compromising performance in downstream tasks. Future directions, possible clinical use cases, and limitations are discussed.