Question about bouyancy... Looking for a hint

Here is a thought experiment that may make the solution clearer.

If you weighed an iron ball in a vacuum would it weigh the same as it weighs at the bottom of a swimming pool? By weight I mean the force the ball would apply to a scale.

In a vacuum the iron ball would weigh
[mass of iron * acceleration due to gravity]
Gravity pulls the ball down and there is no force pushing up.

On the other hand at the bottom of a pool it would weigh
[mass of iron * acceleration due to gravity - mass of displaced water * acceleration due to gravity]
Gravity pulls the ball down but there is now a buoyancy force pushing up on the ball.

Would the iron ball need to sink as far into mercury to support its weight at the bottom of a swimming pool as it would in a vacuum?

edit to improve clarity ???

I've been thinking hard about a good microscopic explanation of Archimede's Law, and why the principle doesn't change with the materials you use. I think you need four things:
(1) A gravitational field. Can't have bouyancy without it. Gravity pulls your solid body down, and creates a density gradient in your fluid ( air or water ) with the highest density at the bottom.
(2) A fluid medium. Fluids can't support shear forces, hence 'hydrostatic' pressure is exerted equally in all directions ( isotropic ) at a given point within the liquid. This does not mean the pressure is the same at all points within the fluid. For most reasonable equations of state, a fluid will increase it's pressure with increasing density ( compositon and temperature constant ).
(3) A solid body. This gives a constant size and shape, that is constant surface areas to present to the surrounding fluid.
(4) Nothing else. No thermal gradients, explosions, adsorptions, absorbtions, etc...
Now any small element of the body's surface facing toward the fluid will have a force exerted on it, which resolves to a normal or perpendicular direction to that surafce ( fluids don't support shear ). This force will point inwards, being balanced/resisted by whatever holds the body in shape, so there's no collapse/expansion. Bouyancy is a upward force which may or may not exceed the weight of the body. This bouyant force is the total of all the perpendicular forces over all of the surface that is in contact with the fluid. This is a vector summation. Any surface elements that have a normal in the horizontal plane ( thus the tangent plane to the surface is aligned with a vertical plane ) will only experience lateral forces, not up or down, and thus are not relevant to bouyancy. Surface elements with a normal directed somewhere below the horizontal plane will experience an upward force from the fluid, which resolves to the component of the force from the fluid in the vertical direction ( the horizontal component not contributing ). Those with a normal directed somewhere above the hozizontal plane will experience a downward force ( components resolved similiarly ). Thus for a sphere, which has a helpful symmetry, the force summation ( nett direction down ) from the upper hemisphere is pushed upon by less dense/pressurised fluid than the force summation ( nett direction up ) from the more dense/pressurised lower hemisphere. Phew!!!
Now if we define ( in our imagination ) some volume, within the greater fluid body, which has the exact outer surface ( say a spherical shell ) as the solid body we are actually using, then we conclude that our solid is displacing a mass = volume*density of that fluid. The surrounding fluid is supporting our real body to the same degree as the volume of fluid it has displaced. If our body has the same density as the fluid, it will remain in situ as did the fluid within our imaginary shell in the absence of the solid. If our body has greater density then we gain no more bouyancy, but we have more mass within that volume, so gravity wins and we sink. If the body is less dense then bouyancy is still unchanged, but we have less mass within the volume, so gravity loses and we rise.
Replacing air by water in the original question thus increases the density of the fluid that our solid is replacing ( above the mercury level ), we will thus have an increase in the bouyant force from that. The sphere was initially sitting to some level upon or partly within the mercury fluid. It was displacing a fraction of the sphere's worth of mercury, and it had reached some equilibrium between gravity ( down ) and bouyancy ( up ) from the mercury prior to adding the water. So you know what happens when we add the water ....... :-)

(edit) I've thus implied some shape dependence here, which could introduce some torque as well.

(edit) I've implied that if there is not a surface element which is in some way downward facing which is in contact with the general fluid body then Archimedes is negated. ( 'Suction' cups used by glaziers to handle glass for instance, or underwater caissons used in construction ).

(edit) 'Eureka' in ancient Greek means 'I need a towel!' ;0)

(edit) The 'swim bladder' of some fish is an internal balloon to which gas is added to, or subtracted from, by chemical means. It changes the volume of water that the fish displaces. Adjust this to suit the water conditions at a given depth for neutral bouyancy on a given day, and you reduce the energy to be expended by swimming to stay at a certain depth.