Is it possible to construct a secure block cipher of size $2n$ given a secure block cipher of size $n$?
One method of doing this would be to construct a Feistel network using the n-bit blockcipher as a round function. A drawback of this approach is that because you're using a blockcipher (a pseudo-random permutation) in place of a pseudo-random function, you are almost immediately limited to being secure to only $q \ll 2^{n/2}$ queries as a result of the Birthday Paradox. For $n = 64$, this could be quite problematic. (And I guess might be the type of scenario that led you to ask this question in the first place).

One way around this is to build a PRF $f_{K_1 K_2}(x) = E_{K_1}(x) \oplus E_{K_2}(x)$, where $E$ is your blockcipher. (See "The Sum of a PRPs is a secure PRF"). This is secure for $q \ll 2^{2n/3}$ queries. The result generalizes to $q \ll 2^{dn/(d+1)}$ when you XOR the outputs of $d$ independently keyed blockciphers. So you could use this for your round function. But since your round functions have to have different keys, you end up using a lot of key material, and there's a significant performance hit.

As an aside, there's a lot of literature on constructing $m$-bit "tweakable" blockciphers out of $n$-bit blockciphers for arbitrary $m &gt; n$. This line of work is motivated mostly by full-disk encryption. See, for example, EME, HEH, HCTR. But almost all of these are again only secure up to $q \ll 2^{n/2}$ queries.