Fortunately as there are KML import/export options for most features, 
so can export the route as KML then reimport it into a My Map. I've 
detailed the export and re-import steps below... 

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Export Driving Directions 

First create your route in Google Maps, and click ‘Link to This Map’ 
and select and copy the top ‘email’ link to the clipboard. 

How, how you continue depends if you have more than one destination. 
If you added more, or dragged the route to customise it then it will 
actually contain many destinations, which the inbuilt KML export cant 
cope with. 

* If only a single destination can just append &output=kml to the end 
of the url you copied, and use it in the import below. 

* however if you have multiple destinations then I have a little 
script to create a KML file that gets round the limitation. 
...1. Goto http://www.nearby.org.uk/multi-to-kml.php 
...2. Paste in the URL you copied earlier, and click Go 
...3. Right click the “Download Google Earth KML File” link and copy 
the shortcut to the clipboard, this is the url you can use in the 
import process 

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Import into My-Map 

Create a new MyMap - or open a previously created one you want to 
import into. 

* Goto Edit mode on the Map 
* Click Import top left of the editor window 
* Paste in the URL you prepared earlier (remember to add &output=kml 
if using the output direct from GoogleMaps) into the ‘enter the url of 
map data on the web’ box. 
* Click Upload 
* Be prepared to rename your map, as often the import will name the 
MyMap as per the file just imported 

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… sounds complicated but not too difficult once you get used it it.