My method is similar to Spin’s but a bit different.
Work in orthagonal mode, front view.
If your two halves don’t meet on a vertical grid line then delete the copy-half in object mode. Then go back into edit mode and move your original half mesh till the open edge (the centre loop) sits almost exactly on a vertical grid line (I try to model on the origin centre line for this reason). Zoom in close if you have to.
Now, place the cursor on the vertical grid line then hit Shift-S- Cursor>Grid. Select the centre loop (should be able to do an edge select with nothing else selecting but that loop). Select the “cursor” pivot point from the pop up menu so that scaling takes place with regards to the cursor you just positioned. Then, hit S, X to scale the selected vertices horizontally a fraction. Move your mouse toward the cursor and watch the readout bottom left of screen till it says (0,0,0) then release.
Now, go back into object mode, alt-D to make your copy. Ctrl-M to mirror it about the X-axis. Now, since you still have the cursor pivot option selected, it should flip exactly across the centre line so the centre loops of each half are exactly aligned. Now CTRL-J to join the meshes.
Go back into Edit mode, hit A to select all vertices then W - Remove Doubles. Ctrl-N - Normals to outside. You’re done. Rotate the model and look for telltale signs of missed vertices like pinches or dark patches (there shouldn’t be any but no one’s perfect). If you find some, join them manually by selecting a pair then S-X and move the mouse till they coincide - then W-Remove double to join them. You could just merge them but this doesn’t guarantee the’ll sit on the centre line.
It sounds complicated but really takes less than a minute no matter how complex the model is, especially if you already model with central grid line. And it becomes a habit very quickly. I often use the delete and re-centre routine (without joining) throughout the modelling process as a way of keeping things tidy.
I also use this same approach when scaling features while building the model. Always try to keep your central vertices centred on a common grid line and life is so much easier.
Practise on half a UV sphere if you’re unsure.