In June it will be hot, hot, hot!
I love Paris - it's expensive though.
Echo what the others said - especially about Montmartre and the Sacre Couer - breathtaking. The Louvre is really overrated I think - if you have an interest in art/art history I'd recommend teh Mussee D'Orsee and the Picasso museum.
We also did a very touristy, but lovely, boat trip down the Seine which was gorgeous, and a really different way to see the city. IIRC we got on the boat at the Notre Dame - which ais a fabulous area in itself - lots of artists, street markets, street performers etc.
Best views of the Eiffel Tower are from Place de la Conchord -its a voew straight across the city and is gorgeous (plus there is a grgous creperie there -
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Ummm, what else? For lunches we used to buy fresh bread/croissants, cheeses and meats and have a little picnic in one of the many little park areas - gorgeous and a good way to keep the costs down a bit.
Ooh, try and see if there is a show on at the Moulin Rouge whilst your there - we went on our last night and it was amazing!
Noah Gabriel 11.06.05 
Elijah Cameron 28.04.06
Sebastian James 15.05.08
Nathaniel Joseph 04.06.10