I was trying to figure out the answer and I found these maps that would suggest that they merged in the 9th century. But then I was reading about various Conans of Brittany who seemed to be independent Dukes? What is the deal with this?
Merovingian dynasty - Merovingian dynasty - Wikipedia
Francia at the death of Pepin of Heristal, 714 - Charlemagne - Wikipedia
Carolingian empire 843 888 - Territorial evolution of France - Wikipedia
You might find a summary for Early Medieval Brittany political history there
As you can see, the question of whenever Brittany was part or not part of the Frankish realm depends of what we mean by that : it was certainly not considered as a "core" part of it as were, say, Austrasia or even Aquitaine, but there are similarities with both the "inner" and "outer" Frankish spheres of influence in expectation of obedience, tribute and military support as could be from Bavarians, Thuringians or Frisians (in various degrees).
Bretons themselves seem to have played the game depending on the situation : periods of royal decline in Francia, as for other peripheral peoples, nevertheless seem to corresponds with periods of renewed ambition and independence, which in Brittany took the form of a successful military resistance, which never went to the point of utterly refusing a thin form of relation.
Indeed, the power, prestige and influence of Francia wasn't necessarily opposite to ambition and independence : Breton petty-king and nobles could even find the means of obtaining just that by being integrated into more global politics, interconnected with the Frankish ensemble, both against their own rivals but as well the Frankish kings themselves.
The "deal with that" is the same than elsewhere in Europe in the same period, how balance of power, 'negotiated' trough military and diplomatic means, saw the emergence (but as many collapses) of emerging entities out of flux and reflux of primary states. In that Early Medieval Brittany was definitely, if on the edge, of the Frankish ensemble even while being wholly politically independent of it.
u/Libertat has already covered the early period; there are some earlier answers by u/mimicofmodes that will help for later periods: a question about Constance of Brittany for the 12th century, and Why was Anne of Brittany so determined to keep her duchy independent from the French crown? for the 15th century.
In short, although Brittany was basically always subject to France (or England...or both at the same time), it was legally independent until...well 1488 at least, and depending on how you look at it, maybe until as late as 1547.
