Union Boolean Modifier
You do not specify for certain in your question if you require the geometry of the two edge loops to match. With this in mind, if you do want the geometries to match, use Leon's Bridge Edge Loops answer. However, if you are ok with the geometeries not matching, then one very easy way to do this is to join the handle and the cup with a Boolean modifier set to Union. This method does not create new faces on the cup joining to each face of the handle. Nonetheless, it still will join the two. Here's how:
If your handle is a Curve object (rather than a mesh object), then convert it to a mesh object:
Select handle → ALTC → Convert to Mesh
Position the handle and the cup together like you'd like them to be joined.
Add a Boolean modifier to the cup:
Select cup → Object Properties window → Modifiers tab → Boolean
For Boolean's options:
Click the Apply button on the Boolean modifier and now the two will be joined:
