One of the key features of sixth generation (6G) mobile communications will be integrated sensing and communication (ISAC). While the main goal of ISAC in standardization efforts is to detect objects, the byproducts of radar operations can be used to enable new services in 6G, such as weather sensing. Even though weather radars are the most prominent technology for weather detection and monitoring, they are expensive and usually neglect areas in close vicinity. To this end, we propose reusing the dense deployment of 6G base stations for weather sensing purposes by detecting and estimating weather conditions. We implement both a classifier and a regressor as a convolutional neural network trained across measurements with varying precipitation rates and wind speeds. We implement our approach in an ISAC proof-of-concept, and conduct a multi-week experiment campaign. Experimental results show that we are able to jointly and accurately classify weather conditions with accuracies of 99.38% and 98.99% for precipitation rate and wind speed, respectively. For estimation, we obtain errors of 1.2 mm/h and 1.5 km/h, for precipitation rate and wind speed, respectively. These findings indicate that weather sensing services can be reliably deployed in 6G ISAC networks, broadening their service portfolio and boosting their market value.