The easy way to do this is simply to sign in to Ubuntu One in your browser, and then put https://one.ubuntu.com in the iframe; you'll stay signed in for a while at least (and you're signed out occasionally for security).
The hard way, if you want to be sure that you're always signed in, is to take the URL https://one.ubuntu.com/api/1.0/from_oauth/?next=/ and sign it with a valid Ubuntu One OAuth token. You can retrieve a token using the Python library in https://launchpad.net/ubuntuone-couch or manually from your Ubuntu keyring. Once you've signed the URL with that OAuth token, serialise the OAuth signature into the URL (so you'll get a URL which looks like
and then open it in your web browser. That URL will take you to Ubuntu One, in the browser, signed in as you.
You may find the following page useful, which does the signing in JavaScript: edit this page to include your token details, and then you can just open this page in the browser and bookmark it, or link to it in an iframe with a file:// URL, and it should work. It also requires oauth.js and sha1.js from http://oauth.googlecode.com/svn/code/javascript/.
<!doctype html>
<html><head><title>Log in to Ubuntu One without password</title>
<script src="sha1.js"></script><script src="oauth.js"></script>
</head><body>
<script>
var url = "https://one.ubuntu.com/api/1.0/from_oauth/"
var accessor = {
token: "TTTTTTTT",
tokenSecret: "SSSSSSSSS",
consumerKey : "CCCCCCCC",
consumerSecret: "ssssssssss"
};
var message = {
action: url,
method: "GET",
parameters: {next: "/"}
};
OAuth.completeRequest(message, accessor);
OAuth.SignatureMethod.sign(message, accessor);
url = url + '?' + OAuth.formEncode(message.parameters);
location.href = url;
</script>
</body>
</html>