Have you tried doing this with syncing your bookmarks over google account?
I know that stock browser reads the bookmarks synced with google account when logged in.
If you log into google using chrome browser, syncing bookmarks and then logging into google using stock browser should get your bookmarks there.
Can't test it because I am currently on 4.2.2 and don't have Chrome for android installed so this is just a guess how it could be done.
When you first start your chrome browser it asks you right away to login into your google account.
If you have done that it asks you the same question as in the desktop version - whether to synchronize things like passwords, bookmarks etc. between your chrome instances.
If you have done the first thing and answered the following question with "yes everything" or just "yes" with a marked checkbox for bookmarks you're pretty much set up.
If you don't have this done yet you can reconfigure synchronization in chrome settings.
Now if you take a look into the bookmarks section of chrome you should see some folders like "mobile bookmarks", "other bookmarks", "desktop bookmarks" (due to my german version the titles may differ a bit).
Every bookmark saved under "desktop bookmarks" will be synchronized back to your google account.
If you open your stock android browser some minutes later and login to your google account when asked you will see your google email address as a folder/category inside bookmarks section containing your bookmarks from chrome.
Even if you will install i.e. a chrome browser on your desktop some day you will end up with your bookmarks/tabs/passwords/... beeing pulled from your google account once you login.
Here are some screenshots (as I already mentioned they're unfortunately from my german version):