There are two FARs that govern takeoff. The first is §91.155 and quite clearly states that no one may take off under VFR (except at Class G airports) if the ceiling is less than 1,000′. You could ask for a Special VFR, but you might have to wait for IFR traffic to land and take off since they have priority. You won’t get it at Class B—in fact many (most?) have a notation on the chart saying that it is not allowed.
The second FAR is §91.175 (f) governs visibility in IFR operations and does not apply to Part 91 operations (which I assume most readers of this post are).
So assuming you want to depart VFR, you would need 1,000′ ceiling to depart, except for an airport in Class G airspace where the takeoff and landing require the same weather conditions as the airspace—1 mile visibility and clear of clouds. If you want to depart IFR, you can do so with 0 ceiling and 0 visibility. (Subject to any conditions in the Obstacle Departure Procedure §91.175 (f)(3)).
§91.155 Basic VFR weather minimums.
(c) Except as provided in §91.157, no person may operate an aircraft beneath the ceiling under VFR within the lateral boundaries of controlled airspace designated to the surface for an airport when the ceiling is less than 1,000 feet.
(d) Except as provided in §91.157 of this part [Special VFR—which you won’t get at a Class B airport], no person may take off or land an aircraft, or enter the traffic pattern of an airport, under VFR, within the lateral boundaries of the surface areas of Class B, Class C, Class D, or Class E airspace designated for an airport—
(1) Unless ground visibility at that airport is at least 3 statute miles; or
(2) If ground visibility is not reported at that airport, unless flight visibility during landing or takeoff, or while operating in the traffic pattern is at least 3 statute miles.
(e) For the purpose of this section, an aircraft operating at the base altitude of a Class E airspace area is considered to be within the airspace directly below that area.
§91.175 Takeoff and landing under IFR.
(f) Civil airport takeoff minimums. This paragraph applies to persons operating an aircraft under part 121, 125, 129, or 135 of this chapter.
(1) Unless otherwise authorized by the FAA, no pilot may takeoff from a civil airport under IFR unless the weather conditions at time of takeoff are at or above the weather minimums for IFR takeoff prescribed for that airport under part 97 of this chapter.
(2) If takeoff weather minimums are not prescribed under part 97 of this chapter for a particular airport, the following weather minimums apply to takeoffs under IFR:
(i) For aircraft, other than helicopters, having two engines or less—1 statute mile visibility.
(ii) For aircraft having more than two engines— 1⁄2 statute mile visibility.
(iii) For helicopters— 1⁄2 statute mile visibility.
(3) Except as provided in paragraph (f)(4) of this section, no pilot may takeoff under IFR from a civil airport having published obstacle departure procedures (ODPs) under part 97 of this chapter for the takeoff runway to be used, unless the pilot uses such ODPs or an alternative procedure or route assigned by air traffic control.