Is it an evergreen?

No. It dies back to the ground each year.

Could it possibly be "Sweet Autumn Clematis"?

Here is a link that might be useful: Terniflora

My clematis is blooming now. I do have a Sweet Autumn Clematis that blooms later in the fall.

However, that picture does look a lot like mine. What do you think? Would a June blooming clematis fit the description?

My handy little book says this could be flammula, Z 7-9, A Mediterranean species with clouds of small, fragrant flowers, grows up to 15ft.

Hope that's it.

Paul Farges and Apple Blossom are close, but I can't be sure from the pictures I have, they are small.

Thanks, virakech. One of your suggestions lead me to this page. Do you think it is a clematis armandii?

Here is a link that might be useful: Clematis armandii

Clematis armandii is evergreen.

Does it have a scent?

It is at the end of its blooming period now. I don't notice a scent.

Does it climb? If not maybe it is an herbaceous type like clematis recta.

Yes! I've done some investigating and I think it is clematis recta. Thanks, everyone!

I have a C. recta blooming now. The flowers are tiny, smaller than a dime. Are yours that small?

Here in zone 5, mine is just finishing its blooming period. The flowers are larger... more like quarter size.

I think it's fargesioides. Look up 'Summer Snow', Paul Farges on Clematis on the Web. Mine just finished blooming in SW Ohio. I'm pretty sure that's what you have there.

I've looked quickly at a couple pictures on the web. It does look like mine. I'll do some further investigating later. Thanks for that suggestion.