I've found the solution:
Close the "App Store" App
Open "Terminal"
Enter defaults write com.apple.appstore ShowDebugMenu -bool true
Open the "App Store" app
Now you have in the menu-item "Debug" - open it - and open "Show Download Folder"
Delete all files and folders in the folder "com.apple.appstore"
Close the "App Store" app
Restart the Mac
Clean the trash bin
Open the "App Store" app an repeat the download.
Now the download should work!
If you want to hide the debug-menu again just enter in Terminal:
defaults write com.apple.appstore ShowDebugMenu -bool false