I disagree with your assessment. It's still a bit of a run-on sentence in this case. I'd just break it into two related sentences because it's comprised of separate but related thoughts.
"The economy has been sluggish for four years now. However, some signs of improvement are finally beginning to show."
Sure - that works fine. But it is a rewriting without a clause. It doesn't change how a clause should be punctuated without the conjunction.
vince's question was:
Would the following be correct?-
"The economy has been sluggish for four years now; some signs of improvement are finally beginning to show." The co-ordinating conjunction 'but' has been removed.
The removal of the conjunction 'but' brings us back to the rule about a linking a dependant clause or phrase to an independant clause:
Do not use a semicolon to link a dependent clause or a phrase to an independent clause.
[WRONG] Although gaining and maintaining a high level of physical fitness takes a good deal of time; the effort pays off in the long run.
Although gaining and maintaining a high level of physical fitness takes a good deal of time, the effort pays off in the long run.
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