How do I write a GeoPandas dataframe into a single file (preferably JSON or GeoPackage)?

To write to GeoJSON: dataframe.to_file("output.json", driver="GeoJSON")

To write to GeoPackage: dataframe.to_file("output.gpkg", driver="GPKG")

Documentation is here, though somewhat sparse.

@toms' answer is perfect -- which I've used for creating my geopackage, btw. Now that I've learned, I want to add to the discussion:

GeoDataFrame's to_file method has the following signature:

>>> help(dataframe.to_file)

to_file(filename, driver='ESRI Shapefile', schema=None, **kwargs)
...

And it follows:

...
A dictionary of supported OGR
    providers is available via:

    >>> import fiona
    >>> fiona.supported_drivers

As of fiona.__version__ == 1.8.6 those are the supported drivers:

{'AeronavFAA': 'r',
 'ARCGEN': 'r',
 'BNA': 'raw',
 'DXF': 'raw',
 'CSV': 'raw',
 'OpenFileGDB': 'r',
 'ESRIJSON': 'r',
 'ESRI Shapefile': 'raw',
 'GeoJSON': 'rw',
 'GPKG': 'rw',
 'GML': 'raw',
 'GPX': 'raw',
 'GPSTrackMaker': 'raw',
 'Idrisi': 'r',
 'MapInfo File': 'raw',
 'DGN': 'raw',
 'S57': 'r',
 'SEGY': 'r',
 'SUA': 'r',
 'TopoJSON': 'r'}

Which is a pretty interesting list, btw. It is just not clear the relation between file-formats extension/name, which may look a bit "noisy" for newcomers (like me), but a second round of googles will do it.