function construction - Force evaluation of the right-hand side of a local variable definition

One can do

expression = a + 1;

f[a_] := Evaluate[expression]

but how would one do something like this

expression = a + 1;

g[x_] := Module[{a,b},

  a=1;
  b=Evaluate[expression];
  x*b
 ]

so that only variable b right-hand side would be evaluated and not the whole Module. This would be useful for using long formulas generated by mathematica in own function definitions.

The solution suitable for me:

Thanks Jens for the answer, that pointed me to the right direction. The solution I was looking for and works for me is

expression = a + 1;


With[{expr=expression},
 f[x_]:=Block[{a,b},
   a = 1;
   b = expr;
   x*b
 ]
]

So using With outside of SetDelayed causes evaluation of expression as desired (Jens answer) and the naming problem (a and $a) is solved by using Block instead of Module.