How to combine Lighttpd simple_vhost with www to non-www redirects

I've set a simple virtual hosting using Lighttpd simple_vhost module and want to do a permanent 301 redirect from www to non-www.

Here are most significant parts of my configuration file:

server.modules = (
  "mod_simple_vhost",
  "mod_redirect",
)

simple-vhost.server-root = "/Websites/"

$HTTP["host"] =~ "^www\.(.*)$" {
  url.redirect  = (
    "^/(.*)" => "http://%1/$1",
  )
}

The redirect block is taken as-is from the Lighttpd documentation, but it doesn't work at all. There is subtle difficulty mentioned on the simple_vhost module docs:

You have to keep in mind that conditionals and simple-vhost interfere
with one another.

I've tried the solution proposed there (http://redmine.lighttpd.net/projects/1/wiki/Docs_ModSimpleVhost), but still without success:

$HTTP["host"] != "www.example.org" {
    simple-vhost.server-root = "/Websites" 
    simple-vhost.default-host = "example.org"  
}

$HTTP["host"] == "www.example.org" {
    server.document-root = "/Websites/example.org" 
}

What is the proper way to do redirects from www to non-www when using virtual hosts?

Here is my full config file.

server.modules = (
"mod_access",
"mod_simple_vhost",
"mod_alias",
"mod_accesslog",
"mod_compress",
"mod_redirect",
"mod_cgi"
)

server.follow-symlink = "enable"

server.document-root = "/Websites"

server.errorlog = "/Websites/error.log"
server.breakagelog = "/Websites/breakage.log"

index-file.names = ("index.html")

accesslog.filename = "/Websites/access.log"

## deny access the file-extensions
#
# ~ is for backupfiles from vi, emacs, joe, ...
# .inc is often used for code includes which should in general not be part
# of the document-root
url.access-deny = ( "~", ".inc" )

##
# which extensions should not be handle via static-file transfer
#
# .php, .pl, .fcgi are most often handled by mod_fastcgi or mod_cgi
static-file.exclude-extensions = ( ".php", ".pl")

server.port = 80
server.bind = "10.20.30.40"

## to help the rc.scripts
server.pid-file = "/var/run/lighttpd.pid"

## virtual directory listings
dir-listing.encoding = "utf-8"
server.dir-listing = "enable"

server.username = "lighttpd"
server.groupname = "lighttpd"

#### compress module
compress.cache-dir = "/var/cache/lighttpd/compress/"
compress.filetype = ("text/plain", "text/html", "application/x-javascript", "text/css")

#### external configuration files
## mimetype mapping
include_shell "/usr/share/lighttpd/create-mime.assign.pl"

## load enabled configuration files,
## read /etc/lighttpd/conf-available/README first
include_shell "/usr/share/lighttpd/include-conf-enabled.pl"

# Simple virtual host.
$HTTP["host"] != "www.example.com" {
simple-vhost.server-root = "/Websites/"
}

$HTTP["host"] == "www.example.com" {
server.document-root = "/Websites/example.com/"
}