Beirut to Alexandria

As part of a large travel plan to circumnavigate the Med. from Cadiz
to Tangier, I plan to go from Beirut to Alexandria. I'm slowly peicing
together parts of the plan like a giant jigsaw, but anyone who can
help with this peice of the jigsaw would be appreciated:

How can I do it? Is there a ferry? Any info. on this ferry (who, what,
where, how much etc). Or are there better (and cheaper!) alternatives.
Despite this being a grand plan i'm trying to do the whole thing on a
budget.

Thanks, TG.