Could be clematis wilt, which can happen for what seems (to me) to be no reason. What I do is to cut away the infected stem/s to ground level, top up the earth round the roots and hope it will send up new shoots. There may be better ways to deal with this, however, if it is wilt.

Looks like the damage is on the one stem only. As Oji says wilt can cause this to happen - but also check back that you haven't damaged that one stem growing up. Being so thin they are easily bruised or damaged when you plant or tie in. This slightest knock or nick can cause this.
I always thought that wilt didn't usually show up until it flowered - I maybe wrong on that one though :)

I planted a clematis a few years back. The 1st two years it suffered with the Clematis wilt, I just cut it back to about 8" from the ground and mulched it in the Autumn and now I don't have a problem, but like you I thought mine was on it's last legs. I hope this helps.

Thank you kindly, Ojibway93, Scottish, and jennyh. Looking closely, the right-hand stem does seem to have been injured, as Scottish suggests. Should I cut it back to slightly below the injury, or must I cut it back almost to ground level?

If there are plent of shoots coming out the ground, I would take it right down - if not take it back to a set of strong leaves. Although I'm not sure if this would be the correct method but it's what I would do Deanna.