What is probably happening here is you have power to the light on one black, another black to the switch, and a return path along the whites. That is, the switch is AFTER the light.
This is generally unsafe as the light is still energized, even when the switch is off, and should be fixed regardless.
The first thing I'd do is beg/borrow/steel a non-contact voltage detector.
With everything still attached, switch the light off. If I'm right, the detector would read hot on BOTH black wires, and the whites will be un-energized. Flip the switch and the whites should be energized as well.
If this is indeed the situation, the fix is relatively easy. Turn the power off! Remove the existing light.
Figure out which pair is wired to the switch. The easiest way is to use a meter in resistance mode. Find the black-white pair that shows low resistance when the switch is on, and open circuit when switched off. Mark the white wire with a loop of electrical tape. This will be the switched hot.
Now wire the blacks together with an appropriate wire nut, and make sure it's a secure connection. You should not be able to see any copper.
Wire the switched hot (the white wire marked with tape) to the black wire of the new fixture. Wire the remaining white to the white wire of the fixture. Make sure you've grounded the fixture according to the manufacturers instructions.
Mount the fixture, and turn the power back on.
Knob and tube is no longer listed and approved, it should be de-energized and abandoned when encountered within the scope of any new work after being replaced with listed cable assemblies suitable for the location (in your case, non-metallic bonded cable, aka Romex).
That said, it sounds like what you have in the box used to be a switch leg where the original knob and tube feed part of the leg is no longer present (hence, the absence of the neutral), and through the process of removing it the lighting circuit was changed so that the leg ended up being always hot. What you should do is just disconnect it from whatever fixture is now back-feeding it. Be gentle, that stuff is brittle and the insulation can just disintegrate in your hands as you handle it.
If that's the case, then you probably don't have a multi-feed (two circuits on the same phase sharing the same neutral) going on, it was just a switch leg, but you should probably consider retiring the rest of the old wiring anyway.
If you can't find where it's being fed, your immediate concern is now to ensure that the box is suitably grounded if it's metallic, you can test this by attaching one end of a voltage tester to any known hot leg, and the other to the box itself. You should read around 120V and be able to confirm the presence of some kind of physical ground connected to the box. If that's not the case, call an electrician and have them straighten it out.
Still, the best guidance in these cases is just get rid of it where / when you see it, especially if it's been disturbed recently. That old feed is probably not going to cause you any problems, but please consider erring on the side of caution and having that old lighting circuit re-fed.
I added two images showing knob and tube wired light fixture and two wire light switch. Romex out of fixture goes to switch.
Best Answer
Actually, you very likely do NOT have a black hot and a white neutral. A simple (not smart, not dimmer, not motion, not 3-way, etc.) switch will have a hot and a switched hot. So the white you see is not neutral.
The 4 wires you need are ground (all grounds go together, everything has a ground, so we'll ignore that for now), neutral (the problem), hot and load. Load is also referred to as switched hot.
There are two possible configurations of your current switch & light:
In this case, you would typically have two cables coming into the box. One would have black hot and white neutral, with black going to the switch and white connected to the white of the other cable. The second cable would have black switched hot and white neutral, with the black switched hot connected to the switch.
Power goes to the light first. At the light, white is neutral and connects to the light. Black is hot and connects to another cable (which also connects switched hot to the other side of the light. The end result is that the black/white cable you have at the switch does not contain neutral.
From your description, I suspect you have the second configuration, and you need to add a wire to get neutral at the switch. The problem is you can't simply "run another wire" - all wires need to be together in conduit (unlikely since if you had conduit then the switched hot should not have been white) or together in a cable, which would require a 3-wire cable instead of a 2-wire cable (ground isn't counted in that numbering scheme).
Current code requires neutral to be included in switch boxes, but that is a relatively recent change so it is quite common to find switch boxes without an accessible neutral.
Neutrals must match their hots - you can't pull another neutral from a different light (or even worse, a different circuit). You also can't reuse ground as neutral (that would technically work but be a significant code violation).