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Most solubility references (e.g. this one) say that there are "few" insoluble salts of alkali metals.

Are there any exceptions?

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Lithium carbonate has poor water solubility. Cesium triphenylcyanoborate is insoluble in water.

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They would advertise $\ce{NaBPh4}$ as a reagent to determine $\ce{K+}$ gravimetrically. Old-timers may remember the feared Kalignost. $\ce{XBPh4-}$, for X=$\ce{Rb+, Cs+, NH4+,Tl+}$ is similarly insoluble.

Thanksfully, now there is methods like AAS, and we plague the kids with that in Instrumental Analysis/Analytical Chemistry classes instead.

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There are some cations whose Group I/ammonium salts are not soluble in water. For example, (thanks Nicolau) the diuranate of ammonium is insoluble in water.

Additionally, (as I just found) the hydrides (H-) of alkali metals are not soluble, it seems, in any room-temperature liquid solvent. They hydrolyze violently (http://goo.gl/Dl8Bd4), and the hydride ion has never been observed in solution.

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    $\begingroup$ I'm not sure whether it's right to say that substances that undergo solvolysis aren't soluble. Technically, they do dissolve, they just react immediately afterwards. $\endgroup$ – Nicolau Saker Neto Feb 22 '14 at 23:34

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