Need to change the owner of an existing SharePoint group using Powershell

Question

I have succesfully been able to create a SharePoint group and assign it an owner. But now I noticed that it only works if the group has not been created. So now I need to add a way to modify the existing group and change the owner. I haven't been succesfull in making it work. Hope some one can shed some light on it. Below is the script I used to create the groups. Thanks.

$SCurl = "http://google.com/sites/sas"

$GroupNamePrefix = "SAS"

$GroupOwner1 = "SharePoint Admins"

$GroupOwner2 = $GroupNamePrefix+" Security Admin"

[System.Reflection.Assembly]::LoadWithPartialName("Microsoft.SharePoint")
$SPSite = New-Object Microsoft.SharePoint.SPSite($SCurl)
$OpenWeb = $SPSite.OpenWeb() 

$OpenWeb.SiteGroups.Add($GroupNamePrefix+" Designers",$OpenWeb.SiteGroups[$GroupOwner1],$OpenWeb.SiteUsers[""],"Members of this group can edit lists, 
document libraries, and pages in the site. Designers can create Master Pages and Page Layouts in the Master Page Gallery and can change the behavior 
and appearance of each site in the site collection by using the master pages and CSS files.")


$OpenWeb.Update()
$OpenWeb.Dispose() 
$SPSite.Dispose()

Answer 1

Here you go. Should be pretty straightforward:

#Get the SPWeb
$web = Get-SPWeb http://url_to_web

#Get the Group
$group = $web.SiteGroups["Group Name"]

#Get the User 
$user = $web.EnsureUser("domain\username")

#Assign that user as the owner
$group.Owner = $user

#Update the Group
$group.Update()

EDIT Added @John Craven's suggestion