Yes, the general welfare clause is all one sentence. Why then do you put your emphasis on the common defence and general welfare, and ignore all that follows the semicolon. That is a list of paticulars qualifying the previous portion of the sentence.
From the Federalist Papers #41:
Some, who have not denied the necessity of the power of taxation, have grounded a very fierce attack against the Constitution, on the language in which it is defined. It has been urged and echoed, that the power "to lay and collect taxes, duties, imposts, and excises, to pay the debts, and provide for the common defense and general welfare of the United States," amounts to an unlimited commission to exercise every power which may be alleged to be necessary for the common defense or general welfare. No stronger proof could be given of the distress under which these writers labor for objections, than their stooping to such a misconstruction.
Had no other enumeration or definition of the powers of the Congress been found in the Constitution, than the general expressions just cited, the authors of the objection might have had some color for it; though it would have been difficult to find a reason for so awkward a form of describing an authority to legislate in all possible cases. A power to destroy the freedom of the press, the trial by jury, or even to regulate the course of descents, or the forms of conveyances, must be very singularly expressed by the terms "to raise money for the general welfare."
But what color can the objection have, when a specification of the objects alluded to by these general terms immediately follows, and is not even separated by a longer pause than a semicolon? If the different parts of the same instrument ought to be so expounded, as to give meaning to every part which will bear it, shall one part of the same sentence be excluded altogether from a share in the meaning; and shall the more doubtful and indefinite terms be retained in their full extent, and the clear and precise expressions be denied any signification whatsoever? For what purpose could the enumeration of particular powers be inserted, if these and all others were meant to be included in the preceding general power? Nothing is more natural nor common than first to use a general phrase, and then to explain and qualify it by a recital of particulars. But the idea of an enumeration of particulars which neither explain nor qualify the general meaning, and can have no other effect than to confound and mislead, is an absurdity, which, as we are reduced to the dilemma of charging either on the authors of the objection or on the authors of the Constitution, we must take the liberty of supposing, had not its origin with the latter.
There's no need to quote the Federal Papers, as they were never agreed upon, debated, vetted through a compromisation process, or ratified into written law by enough states to make it into our Constitution.
At best, they are mere pieces of opinion from a *minority* of Founding Fathers.