What It Is
The British call it “clematis wilt”, though some American gardening books refer to it as “clematis leaf and stem spot”. I prefer to call it “wilt” because it’s very descriptive of the problem! Fortunately the fungus does not attack the root system, so with care your clematis can return to health.
Clematis wilt is a fungal disease formerly called Ascochyta clematidina, now known as Phoma clematidina. It spreads by spores and is helped along by damp or humid weather. The fungus cuts off the plant’s circulatory system so no water can move through its veins, thus causing the wilt symptoms. The fungus can enter the plant through weakened or damaged stems, or can splash onto stems during watering or heavy rain. Plants that are tangled or that stay wet are even more prone to the fungus. Normally wilt appears in the early to mid-summer when the clematis is growing quickly. On an older plant, the woody portion near the ground is often the first place you will notice wilt.
The dramatically beautiful larger-flowered varieties are, sadly, the most prone to the fungus. Species clematis and the smaller-flowered varieties, such as C. montana, C. macropetala, C. alpina and C. viticella are more resistant to wilt.
Wilted leaves on mature Clematis 'Elsa Spaeth'
This young Clematis 'Piilu' was so damaged by wilt that I cut it
completely to the soil line. It sent out new growth within two weeks.
How To Treat It
At the first sign of wilt, cut the affected stems as close to the ground as possible. Be careful not to injure the healthy stems, especially those near the bottom. Remove any fallen leaves. Needless to say, you shouldn’t put these cuttings on your compost. Keep the roots watered, even after cutting out dead foliage. The fungus can overwinter in the dead foliage, so no matter which type of clematis you have, it is recommended that you remove all remaining growth in the fall. Dipping pruners into a weak bleach solution after each cut will prevent spread of the fungus to other plants. Following good sanitation procedures such as these will remove much of the infectious fungal spores and minimize any future problems. Denise Corkery, senior writer at Chicago Botanic Gardens, does not recommend chemical controls such as fungicide be used as treatment for clematis wilt. 
How to Prevent It
It may be impossible to completely prevent clematis wilt. But as with any garden plant, providing the best culture possible will keep your clematis healthy and less susceptible to disease.
1. Site your clematis properly. In the wild, clematis vines scramble over other plants and rocks as they reach for the sun, while their roots remain cool and shaded. You can simulate this natural environment by shading the base of the vine with other plants, a mulch and/or a flat stone. Just make sure not to mulch directly around the stem.
2. Plant deep. Don’t be afraid to plant two sets of leaves under the soil. This will ensure that your clematis grows plenty of underground buds, so even if you must cut affected stems to the ground, the plant will be able to regenerate itself.
3. Provide the right culture. Clematis like a well-drained, evenly moist soil. If your soil is sandy, amend it with peat moss or compost; if clay soil, add compost before planting. Clematis are big feeders, so use an organic fertilizer to keep the soil fertile and rich. In early summer when the plant is full of leaves and buds, keep the soil watered regularly, but not to the point that it is soggy.
4. Furnish support. Clematis stems are brittle and can be damaged by wind. You can help prevent injury by ensuring that your clematis is strongly secured to its support and has plenty of places to attach itself. Train new growth up to the support, manually attaching stems if needed.
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photo credit: Clematis 'Dr. Ruppel' by msanjelpie