Under the plan, the House of Representatives would approve the Senate bill by "deeming" it to have passed as part of a separate measure governing the rules of a House debate on a follow-up health care measure. That follow-up legislation will be designed to change certain controversial portions of the Senate-passed version, a path more palatable to House Democrats who fear that voting for the Senate bill could backfire against them in elections this fall.

Here's the shorter version of the trick: the House Rules committee would declare the Senate bill already passed the House.  And since under the Constitution the House can "determine the Rules of its Proceedings", its declaration stands and the bill can go to the President and signed into law.  For details, you can read it in the WSJ here.  This way the House can send the Senate bill to the President without voting for it.

This is a neat trick, but I don't think the House needs to be this sneaky.  A Constitutional textualist would say the Senate HCR bill (HR3590) can be sent to the President right now and signed into law.  Why is that?  Let's look at the facts.

The fact is that the HCR bill (HR3590) already passed both houses of Congress.  We know HR3590 passed the House - that's why it's an HR bill.  Also, we know the Senate passed HR3590.  So, HR3590 passed both the Senate and the House.  But then, as the WSJ article points out, the Constitution says:

According to Article I, Section 7, in order for a "Bill" to "become a Law," it "shall have passed the House of Representatives and the Senate" and be "presented to the President of the United States" for signature or veto."

But since the bill HR3590 has passed both the Senate and the House, Article I, Section 7 says we can send HR3590 to the President to be signed into law.  QED.