How can web search be disabled in the Windows 10 Anniversary Update (Build 1607)?

To disable Cortana, start regedit.exe, go to HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\Windows Search, create a new 32-bit DWORD value here named AllowCortana and leave its value data as 0.

After a reboot you see the Cortana Logo is gone and the local search opens:

if this also doesn't disable web search, set the value BingSearchEnabled to 0 under HKEY_Local_Machine\SOFTWARE\Policies\Microsoft\Windows\Windows Search

Windows Registry Editor Version 5.00

[HKEY_Local_Machine\SOFTWARE\Policies\Microsoft\Windows\Windows Search]
"AllowCortana"=dword:00000000
"BingSearchEnabled"=dword:00000000

The accepted answer is wrong: it disables Cortana but does nothing to prevent web searches.

According to http://www.thewindowsclub.com/disable-web-search-in-windows-10 you need to do the following:

1. Run "gpedit.msc"
2. Navigate to "Computer Configuration -> Administrative Templates -> Windows Components -> Search"
3. Double click "Do not allow web search"
4. Select "Enabled", click "OK".
5. Repeat step 4 for "Don't search the web or display web results in Search"

I've personally tested this and it worked for me.

NOTE: This will only work on Windows 10 Pro edition. Home edition does not support "gpedit.msc".