Weatherlib sole dependence is JMetarParser, a parsing library used to interpret METAR strings provided by weather observation stations. Thus JMetarParser is at the base of the main task executed by weatherlib.
Anyway, JMetarParser is already included in weatherlib package, whithin lib directory, so just look for the file named jmetarparser.jar and add it to your classpath.
In addition, weatherlib's source code follows the Java 5 specifications, so be sure to run it with JRE 1.5 or any later version.
