In federated learning (FL), a set of participants share updates computed on their local data with an aggregator server that combines updates into a global model. However, reconciling accuracy with privacy and security is a challenge to FL. On the one hand, good updates sent by honest participants may reveal their private local information, whereas poisoned updates sent by malicious participants may compromise the model's availability and/or integrity. On the other hand, enhancing privacy via update distortion damages accuracy, whereas doing so via update aggregation damages security because it does not allow the server to filter out individual poisoned updates. To tackle the accuracy-privacy-security conflict, we propose fragmented FL (FFL), in which participants randomly exchange and mix fragments of their updates before sending them to the server. To achieve privacy, we design a lightweight protocol that allows participants to privately exchange and mix encrypted fragments of their updates so that the server can neither obtain individual updates nor link them to their originators. To achieve security, we design a reputation-based defense tailored for FFL that builds trust in participants and their mixed updates based on the quality of the fragments they exchange and the mixed updates they send. Since the exchanged fragments' parameters keep their original coordinates and attackers can be neutralized, the server can correctly reconstruct a global model from the received mixed updates without accuracy loss. Experiments on four real data sets show that FFL can prevent semi-honest servers from mounting privacy attacks, can effectively counter-poisoning attacks, and can keep the accuracy of the global model.