Is there a web site on which one can type in an English phrase or
sentence (not just a word) and get a free and immediate translation
into Latin?

Dear Nautico, 

Intertran offers a free translator of sites or sentences/phrases into
Latin and from Latin:

<http://intertran.tranexp.com/?> 

The problem with these free machine translations is that they are far
from accurate, and with a language like Latin, it might be a problem.

Take: "cogito ergo sum" - I think therefore I am. 
Intertran translates it as "to think on account of to be"
If I translate it back into Latin it goes: 
"reor propter futurus" 
and so on... 

This is a well known phenomenon in machine translations. Someone once
gave a machine to translate the sentence:
"The spirit is willing, but the flesh is weak" into Russian and then
back to English. The machine answered:
"The vodka was good, but the meat was rotten." 
(In Latin, Intertran answers: 
ghost is voluntary nothwithstanding viscera is pale). 

See on translating Latin for tattoos in LiveJournal: 
http://www.livejournal.com/community/latin/75978.html
 
And here's some more Latin fun from LiveJournal: 
a musical interlude 
<http://www.livejournal.com/users/quislibet/164084.html> 
(A translation to Latin and back of Sir Mixalot's "Baby's Got Back". 

I hope I answered your question. Please contact me if you need any
further clarification on this answer before you rate it.

Many thanks! That's exactly what I was looking for, though I'm acutely
aware of the current limitations of such programs. I have one called
MultiTranse, and I've been testing its ability to render English into
proper Norwegian with relatives in Norway. Amazingly, the responses
I've gotten have been favorable. Someday such programs will work
flawlessly.

Nautico - 

On a related note, here's another reason why not to have tattoos in
languages you can't read:

Disgruntled Japanese Tattoo Artist Inks His Revenge 
<http://forum.japantoday.com/Disgruntled_Japanese_Tattoo_Artist_Inks_His_Revenge/m_77533/tm.htm>

Just used that online program to translate the following into Latin:

History: if you can't sell it, what good is it?

The translation converted back to English: History: if you wish you
can't drive out this, anyone good is this?

What a hoot!

Dear Nautico, 

Thank you for the rating and the tip. I found out in the meanwhile,
that Intertran have a possibility to amend the translation, found
above your results.

I played around with that amending feature, but it's anything but user friendly.