Same-sex marriage approved in the UK

I know that the PM is also a member of parliament (Always in Commons) and has the ability to introduce/vote on legislation.

Commons is one House of Parliament . It goes to the House of Lords next where they will either pass/reject it or place a hold on it for further debate (much like the US Senate). While I'm confident that it will ultimately become law, the House of Lords is the place that it would most likely be tied up due to the large concentration if traditionalists in that house. A good majority of lords are also prominent members of the Church of England.

Thanks for the quick lesson! If the bill passes the House of Lords, does it then immediately become law, or does it need a final signature/approval?