Short answer. You may use the water while it runs through your land, but only while it is there. Article 507 of the Civil Code lets the owner of land where a spring or brook rises use its waters as they pass through the property, but once the water leaves the land it becomes public.
What the law says
The owner of a piece of land on which a spring or brook rises, be it continuous or intermittent, may use its waters while they run through the same, but after the waters leave the land they shall become public
Civil Code, Article 507 — Spring Or Brook On Private Land. Read the full provision →
You may use the water while it is on your land
Article 507 gives the landowner a genuine right to use the water: the owner of a piece of land on which a spring or brook rises, be it continuous or intermittent, may use its waters while they run through the same. It does not matter whether the spring flows constantly or only from time to time — either way, the right to use the water while it is running through your property belongs to you as owner, without needing anyone else's permission for that use.
The right ends where your land ends
The article draws a firm boundary on that right: but after the waters leave the land they shall become public. So your entitlement to the water is tied to its physical location, not to the fact that it originated on your property. Once the flow crosses your boundary and moves onto other land, it stops being something you control and becomes public, regardless of the fact that it rose from a spring on ground you own.
What governs the water once it becomes public
Article 507 states that use of the water, once it leaves your land, shall be governed by the Special Law of Waters of August 3, 1866, and by the Irrigation Law. This tells you that the article does not attempt to regulate the water's use beyond your own property boundary — it hands that off to separate bodies of law dealing with water use generally. What you can rely on this article for is specifically the portion of the flow still within your land.
What this means for using the full flow
So the answer to whether you can use "all" of the spring's water depends on where that use happens. Within your own property, the article gives you the right to use the water as it runs through, without a stated limit on quantity. Once it flows beyond your boundary, that water is no longer yours to control, and any downstream use is governed by the separate water-law framework the article points to rather than by your ownership of the land where the spring originates.